1284 lines
84 KiB
JavaScript
Executable File
1284 lines
84 KiB
JavaScript
Executable File
var ptx_lunr_search_style = "textbook";
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var ptx_lunr_docs = [
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{
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"id": "syllabus",
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"level": "1",
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"url": "syllabus.html",
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"type": "Section",
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"number": "",
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"title": "Syllabus",
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"body": " Syllabus Course Information Day and Time Lectures: TR 11:40 am - 1:20 pm Gladfelter L024 Recitation Section 1: M 12:00 pm - 12:50 pm Wachman 9 Section 2: W 12:00 pm - 12:50 pm Wachman 15 Section 3: F 11:00 am - 11:50 am Wachman 408 Section 4: F 1:00 pm - 1:50 pm Wachman 408 Office Hours See times and location on Canvas Course Goals To introduce some of the basic concepts and techniques of probability and statistics, as applied in the Life and Environmental Sciences. Topics Covered Probability theory and descriptive statistics, probability models, introduction to inferential statistics. Prerequisite MATH 1041 (C), MATH 1038 (C), MATH 1941 (C), MATH 1951 (C), MATH 2043 to 3080 (C-), MA06 Y. Course Mode In-person Credits 4 Textbook and Other Required Materials Textbook Course notes are freely available online . Calculator You will need a scientific calculator (NOT a graphing calculator and NOT a calculator app on your phone\/watch). Any calculator capable of symbolic differentiation\/integration or performing statistical tests is NOT allowed. Scientific calculators (such as the TI-30XIIS ) typically cost around $10. Phone You will need a device (phone or tablet) to take pictures of your quizzes\/tests and upload your work. Assessments Homework There will be homework assignments posted on Canvas approximately weekly. There are more practice problems available in the course notes. Doing homework problems is the primary way that you will learn and internalize the material seen in class. Plan to spend a substantial amount of time throughout the semester doing homework problems. Quizzes There will be in-class quizzes every other week (on Jan 20, Feb 3, Mar 10, Mar 24, and Apr 21). Quizzes give you some practice in a test-like environment without the same grade pressure as a test. The quizzes should be a signal to you about whether you're keeping up with the pace of the course. You may use an allowed calculator during quizzes. Exams This course will have two midterm exams and a Final Exam. Exams are the main opportunity for you to demonstrate your understanding of course material. You may use an allowed calculator and a 3 in x 5 in index card with prepared notes during exams. You are allowed to write on both sides of the index card. You should include your full name in the top right corner of your index card. Your index card will be turned in along with your exam. Test 1 Tuesday, February 17 (in class) Test 2 Tuesday, April 7 (in class) Final Exam Tuesday, May 5, 10:30 am - 12:30 pm Makeup Policy There are no quiz or exam makeups. The lowest quiz will be dropped and replaced with the recitation score. Grading The grade components and their weights are: Component Weight Homework 5% Quizzes 15% Exam 1 24% Exam 2 24% Final Exam 32% The lower cutoff for each letter grade is given in the table below. Letter Grade Lower Cutoff A 93% A- 90% B+ 87% B 83% B- 80% C+ 77% C 73% C- 70% D+ 65% D 55% F 0% Attendance Policy Attendance is expected and will be recorded. If you need to miss a class for some reason, it is your responsibility to get notes from another student for any material you've missed, and to arrange to turn in\/make up any in-class or collected work. Attendance and Your Health To achieve course learning goals, students must attend and participate in classes, according to the course requirements. However, if you have tested positive for or are experiencing symptoms of a contagious illness, you should not come to campus or attend in-person classes or activities. It is the student’s responsibility to contact me to create a plan for participation and engagement in the course as soon as you are able to do so, and to make a plan to complete all assignments in a timely fashion. Expectations for Class Conduct It is important to foster a respectful and productive learning environment that includes all students in our diverse community of learners. Our differences, some of which are outlined in the University's nondiscrimination statement, will add richness to this learning experience. Therefore, all opinions and experiences, no matter how different or controversial they may be perceived, must be respected in the tolerant spirit of academic discourse. Disability Statement Any student who has a need for accommodations based on the impact of a documented disability or medical condition should contact Disability Resources and Services (DRS) in Howard Gittis Student Center South, Rm 420 ( drs@temple.edu ; 215-204-1280) to request accommodations and learn more about the resources available to you. If you have a DRS accommodation letter to share with me, or you would like to discuss your accommodations, please contact me as soon as practical. I will work with you and with DRS to coordinate reasonable accommodations for all students with documented disabilities. All discussions related to your accommodations will be confidential. Academic Freedom Freedom to teach and freedom to learn are inseparable facets of academic freedom. The University has adopted a policy on Student and Faculty Academic Rights and Responsibilities (Policy # 03.70.02) which can be accessed here . Add\/Drop Policy Students will be charged for a course unless dropped by the Drop\/Add deadline date. Check the University calendar for exact dates. During the Drop\/Add period, students may drop a course with no record of the class appearing on their transcript. Students are not financially responsible for any courses dropped during this period. In the following weeks prior to or on the withdrawal date students may withdraw from a course with the grade of \"W\" appearing on their transcript. After the withdrawal date students may not withdraw from courses. Check the University calendar for exact dates. See the full policy by clicking here . AI Policy The use of generative AI tools (such as ChatGPT, DALL-E, etc.) is not permitted in this class unless specifically announced for a particular assignment; therefore, any use of AI tools for work in this class may be considered a violation of Temple University's Academic Honesty policy and Student Conduct Code, since the work is not your own. The use of unauthorized AI tools will result in a grade of zero on the assignment; a second offense will be reported to the Student Conduct Board. Incomplete Grades The grade \"I\" (an \"incomplete\") is only given if students cannot complete the course work due to circumstances beyond their control. It is necessary for the student to have completed the majority of the course work with a passing average and to sign an incomplete contract which clearly states what is left for the student to do and the deadline by which the work must be completed. The incomplete contract must also include a default grade that will be used in case the \"I\" grade is not resolved by the agreed deadline. See the full policy by clicking here . Student Support Services The following academic support services are available to students: The Math Consulting Center Student Success Center University Libraries Undergraduate Research Support Career Center Tuttleman Counseling Services Disability Resources and Services If you are experiencing food insecurity or financial struggles, Temple provides resources and support. Notably, the Temple University Cherry Pantry and the Temple University Emergency Student Aid Program are in operation as well as a variety of resources from the Division of Student Affairs. "
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},
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{
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"id": "notes-01-13",
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"level": "1",
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"url": "notes-01-13.html",
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"type": "Section",
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"number": "",
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"title": "Tuesday, Jan 13",
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"body": " Tuesday, Jan 13 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Sec 1.1: Sets In prior math courses, you mostly asked deterministic questions. Now, we need new tools to model randomness . A toxin molecule in a cell has a certain chance each minute to leave the cell. A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result. The sample space , often denoted , is the set of all possible results of an experiment. A single result is called an outcome , while a collection of results is called an event . An experiment consists of rolling a standard 6-sided die (D6). The sample space is . One possible event is , i.e., the event that the result of the roll is even. Note: we'll use notation like D6 to indicate a 6-sided die with faces 1, 2, 3, 4, 5, 6. Similarly, for example, D4 will indicate a 4-sided die with faces 1, 2, 3, 4. The symbol means \"is an element of\", as in . The symbol means \"is a subset of\", as in . This means that every element of the set is also an element of the set . Consider sets and , each contained inside . We can combine sets in a variety of ways: Union The union of and is the set . Intersection The intersection of and is the set . Difference The set difference is the set . Complement The complement of is the set . Empty Set The empty set , usually written or , is the set which contains no elements. It's useful sometimes to draw pictures called Venn diagrams representing set interactions. Union Venn diagram showing sets with the region representing shaded. Intersection Venn diagram showing sets with the region representing shaded. Difference Venn diagram showing sets with the region representing shaded. Complement Venn diagram showing set with the region representing shaded. Sec 1.2: Probability Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events. An experiment consists of rolling a D6. The sample space is . We might assign probabilities as follows: Distribution for a fair die 1 2 3 4 5 6 Note that we don't have to assign the same probability to each outcome. If we do, we call this distribution uniform . If we have some event, such as , then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event: Note that this is the same result that we would get if we counted the total number of outcomes in and divided by the total number of outcomes in . Given a finite set , the cardinality of , written , is the number of elements in . If is a finite probability space with the uniform distribution and is an event, then: Suppose we have a weighted die that's much more likely to come up 6 than any other outcome. Distribution for a fair die 1 0.1 2 0.1 3 0.1 4 0.2 5 0.1 6 0.4 With this distribution, if , then: Pick a number from 1 to 10. What is the probability of picking 3? Some considerations: Are we using the uniform distribution? What even is the sample space here? Do we need to pick only integers, or is an outcome here? What would \"uniform\" mean if the sample space was infinite? A probability distribution on a sample space assigns probabilities to every event, satisfying the following conditions: . for any event . If , then . Suppose we flip a coin until we see heads. The sample space is . What would \"uniform\" mean here? Is there any way we could assign the same probability to each individual outcome here? Instead of \"uniform\", what if we want to treat the coin as a fair coin? Then what would the distribution be? Distribution assuming a fair coin # flips 1 2 3 4 One last (very useful!) observation. Recall: If , then . For any event , and . So: We can rewrite this in two useful ways: "
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},
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{
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"id": "subsec-Sets-2",
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"level": "2",
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"url": "notes-01-13.html#subsec-Sets-2",
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"type": "Paragraph (with a defined term)",
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"number": "",
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"title": "",
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"body": "deterministic randomness "
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},
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{
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"id": "subsec-Sets-3",
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"level": "2",
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"url": "notes-01-13.html#subsec-Sets-3",
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"type": "Example",
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"number": "1",
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||
"title": "",
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"body": " A toxin molecule in a cell has a certain chance each minute to leave the cell. "
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},
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{
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"id": "subsec-Sets-4",
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"level": "2",
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"url": "notes-01-13.html#subsec-Sets-4",
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||
"type": "Example",
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||
"number": "2",
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||
"title": "",
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||
"body": " A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result. "
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||
},
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||
{
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||
"id": "def-sample-space",
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||
"level": "2",
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||
"url": "notes-01-13.html#def-sample-space",
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||
"type": "Definition",
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||
"number": "3",
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||
"title": "",
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||
"body": " The sample space , often denoted , is the set of all possible results of an experiment. A single result is called an outcome , while a collection of results is called an event . "
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||
},
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||
{
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||
"id": "example-sample-space",
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||
"level": "2",
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||
"url": "notes-01-13.html#example-sample-space",
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||
"type": "Example",
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||
"number": "4",
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||
"title": "",
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||
"body": " An experiment consists of rolling a standard 6-sided die (D6). The sample space is . One possible event is , i.e., the event that the result of the roll is even. "
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||
},
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||
{
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||
"id": "def-subset",
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"level": "2",
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||
"url": "notes-01-13.html#def-subset",
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"type": "Definition",
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||
"number": "5",
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||
"title": "",
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||
"body": " The symbol means \"is an element of\", as in . The symbol means \"is a subset of\", as in . This means that every element of the set is also an element of the set . "
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||
},
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{
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||
"id": "def-set-operations",
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"level": "2",
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||
"url": "notes-01-13.html#def-set-operations",
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||
"type": "Definition",
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"number": "6",
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||
"title": "",
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"body": " Consider sets and , each contained inside . We can combine sets in a variety of ways: Union The union of and is the set . Intersection The intersection of and is the set . Difference The set difference is the set . Complement The complement of is the set . Empty Set The empty set , usually written or , is the set which contains no elements. "
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||
},
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||
{
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||
"id": "subsec-Sets-10",
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||
"level": "2",
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||
"url": "notes-01-13.html#subsec-Sets-10",
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||
"type": "Paragraph (with a defined term)",
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||
"number": "",
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||
"title": "",
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||
"body": "Venn diagrams "
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||
},
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||
{
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||
"id": "fig-Venn-diagram-union",
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||
"level": "2",
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||
"url": "notes-01-13.html#fig-Venn-diagram-union",
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||
"type": "Figure",
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||
"number": "7",
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||
"title": "",
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||
"body": " Union Venn diagram showing sets with the region representing shaded. "
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||
},
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||
{
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||
"id": "fig-Venn-diagram-intersection",
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||
"level": "2",
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||
"url": "notes-01-13.html#fig-Venn-diagram-intersection",
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||
"type": "Figure",
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||
"number": "8",
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||
"title": "",
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"body": " Intersection Venn diagram showing sets with the region representing shaded. "
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||
},
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{
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||
"id": "fig-Venn-diagram-difference",
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||
"level": "2",
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||
"url": "notes-01-13.html#fig-Venn-diagram-difference",
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||
"type": "Figure",
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||
"number": "9",
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||
"title": "",
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||
"body": " Difference Venn diagram showing sets with the region representing shaded. "
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||
},
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{
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||
"id": "fig-Venn-diagram-complement",
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||
"level": "2",
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||
"url": "notes-01-13.html#fig-Venn-diagram-complement",
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"type": "Figure",
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||
"number": "10",
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||
"title": "",
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||
"body": " Complement Venn diagram showing set with the region representing shaded. "
|
||
},
|
||
{
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||
"id": "subsec-Probability-3",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#subsec-Probability-3",
|
||
"type": "Example",
|
||
"number": "11",
|
||
"title": "",
|
||
"body": " An experiment consists of rolling a D6. The sample space is . We might assign probabilities as follows: Distribution for a fair die 1 2 3 4 5 6 Note that we don't have to assign the same probability to each outcome. If we do, we call this distribution uniform . If we have some event, such as , then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event: Note that this is the same result that we would get if we counted the total number of outcomes in and divided by the total number of outcomes in . "
|
||
},
|
||
{
|
||
"id": "def-cardinality",
|
||
"level": "2",
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||
"url": "notes-01-13.html#def-cardinality",
|
||
"type": "Definition",
|
||
"number": "13",
|
||
"title": "",
|
||
"body": " Given a finite set , the cardinality of , written , is the number of elements in . "
|
||
},
|
||
{
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||
"id": "subsec-Probability-5",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#subsec-Probability-5",
|
||
"type": "Fact",
|
||
"number": "14",
|
||
"title": "",
|
||
"body": " If is a finite probability space with the uniform distribution and is an event, then: "
|
||
},
|
||
{
|
||
"id": "subsec-Probability-6",
|
||
"level": "2",
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||
"url": "notes-01-13.html#subsec-Probability-6",
|
||
"type": "Example",
|
||
"number": "15",
|
||
"title": "",
|
||
"body": " Suppose we have a weighted die that's much more likely to come up 6 than any other outcome. Distribution for a fair die 1 0.1 2 0.1 3 0.1 4 0.2 5 0.1 6 0.4 With this distribution, if , then: "
|
||
},
|
||
{
|
||
"id": "subsec-Probability-7",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#subsec-Probability-7",
|
||
"type": "Example",
|
||
"number": "17",
|
||
"title": "",
|
||
"body": " Pick a number from 1 to 10. What is the probability of picking 3? Some considerations: Are we using the uniform distribution? What even is the sample space here? Do we need to pick only integers, or is an outcome here? What would \"uniform\" mean if the sample space was infinite? "
|
||
},
|
||
{
|
||
"id": "def-probability-distribution",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#def-probability-distribution",
|
||
"type": "Definition",
|
||
"number": "18",
|
||
"title": "",
|
||
"body": " A probability distribution on a sample space assigns probabilities to every event, satisfying the following conditions: . for any event . If , then . "
|
||
},
|
||
{
|
||
"id": "subsec-Probability-9",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#subsec-Probability-9",
|
||
"type": "Example",
|
||
"number": "19",
|
||
"title": "",
|
||
"body": " Suppose we flip a coin until we see heads. The sample space is . What would \"uniform\" mean here? Is there any way we could assign the same probability to each individual outcome here? Instead of \"uniform\", what if we want to treat the coin as a fair coin? Then what would the distribution be? Distribution assuming a fair coin # flips 1 2 3 4 "
|
||
},
|
||
{
|
||
"id": "subsec-Probability-10",
|
||
"level": "2",
|
||
"url": "notes-01-13.html#subsec-Probability-10",
|
||
"type": "Example",
|
||
"number": "21",
|
||
"title": "",
|
||
"body": " One last (very useful!) observation. Recall: If , then . For any event , and . So: We can rewrite this in two useful ways: "
|
||
},
|
||
{
|
||
"id": "notes-01-15",
|
||
"level": "1",
|
||
"url": "notes-01-15.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Thursday, Jan 15",
|
||
"body": " Thursday, Jan 15 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Sec 1.3: Conditional Probability Question: How does evidence (e.g., knowledge of one event occurring) change our knowledge of probabilities for other events? Roll a fair D6 two times. Let and . feels more likely if we already know has occurred. The conditional probability of given is: , tells the proportion of which is overlapped by . Two overlapping circles representing events and sit inside a rectangle representing the sample space . The circle labeled is shaded. The portion of that circle which is overlapped by the circle is also filled in with slanted lines. Continuing from the previous example, . So . Then: Notice that is significantly larger than . Diagnostic Testing Setup: A patient takes a diagnostic test. Let be the event that they test positive. Let be the event that they have the disease. The sensitivity of a diagnostic test is . The specificity of a diagnostic test is . But, what the patient really wants to know is . A disease has a prevalence of 1%. A test has sensitivity of 90% and specificity of 91%. For a patient who gets a positive test result, what is the probability that they have the disease? C! Bayes' Theorem (v1) For events with nonzero probability: For example, if a patient sees a positive diagnostic test result, they might try to calculate: is the sensitivity. could be the prevalence. We don't have direct access to . Observation: , so . Observation 2: So: Bayes' Theorem (v2) Continuing from the previous example: What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease? Sec 1.4: Independent Events Question: is supposed to capture how information about affects the probability of . What if it doesn't? Events are independent if . Observation: If have nonzero probability and are independent, then: We can take this last equation as a definition of independence. Continuing , recall and . We found that , so and are not independent. Now consider the event . We have: Therefore events are independent. "
|
||
},
|
||
{
|
||
"id": "subsec-Conditional-Probability-3",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#subsec-Conditional-Probability-3",
|
||
"type": "Example",
|
||
"number": "22",
|
||
"title": "",
|
||
"body": " Roll a fair D6 two times. Let and . feels more likely if we already know has occurred. "
|
||
},
|
||
{
|
||
"id": "def-conditional-probability",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#def-conditional-probability",
|
||
"type": "Definition",
|
||
"number": "23",
|
||
"title": "",
|
||
"body": " The conditional probability of given is: , tells the proportion of which is overlapped by . Two overlapping circles representing events and sit inside a rectangle representing the sample space . The circle labeled is shaded. The portion of that circle which is overlapped by the circle is also filled in with slanted lines. "
|
||
},
|
||
{
|
||
"id": "example-rolls-conditional",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#example-rolls-conditional",
|
||
"type": "Example",
|
||
"number": "25",
|
||
"title": "",
|
||
"body": " Continuing from the previous example, . So . Then: Notice that is significantly larger than . "
|
||
},
|
||
{
|
||
"id": "def-sensitivity-specificity",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#def-sensitivity-specificity",
|
||
"type": "Definition",
|
||
"number": "26",
|
||
"title": "",
|
||
"body": " The sensitivity of a diagnostic test is . The specificity of a diagnostic test is . "
|
||
},
|
||
{
|
||
"id": "subsec-Diagnostic-Testing-5",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#subsec-Diagnostic-Testing-5",
|
||
"type": "Example",
|
||
"number": "27",
|
||
"title": "",
|
||
"body": " A disease has a prevalence of 1%. A test has sensitivity of 90% and specificity of 91%. For a patient who gets a positive test result, what is the probability that they have the disease? C! "
|
||
},
|
||
{
|
||
"id": "thm-Bayes-v1",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#thm-Bayes-v1",
|
||
"type": "Theorem",
|
||
"number": "28",
|
||
"title": "Bayes’ Theorem (v1).",
|
||
"body": " Bayes' Theorem (v1) For events with nonzero probability: "
|
||
},
|
||
{
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||
"id": "fig-P-breakdown",
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||
"level": "2",
|
||
"url": "notes-01-15.html#fig-P-breakdown",
|
||
"type": "Figure",
|
||
"number": "29",
|
||
"title": "",
|
||
"body": " "
|
||
},
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||
{
|
||
"id": "thm-Bayes-v2",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#thm-Bayes-v2",
|
||
"type": "Theorem",
|
||
"number": "30",
|
||
"title": "Bayes’ Theorem (v2).",
|
||
"body": " Bayes' Theorem (v2) "
|
||
},
|
||
{
|
||
"id": "subsec-Diagnostic-Testing-12",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#subsec-Diagnostic-Testing-12",
|
||
"type": "Example",
|
||
"number": "31",
|
||
"title": "",
|
||
"body": " Continuing from the previous example: What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease? "
|
||
},
|
||
{
|
||
"id": "def-independent-events",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#def-independent-events",
|
||
"type": "Definition",
|
||
"number": "32",
|
||
"title": "",
|
||
"body": " Events are independent if . "
|
||
},
|
||
{
|
||
"id": "example-rolls-independent",
|
||
"level": "2",
|
||
"url": "notes-01-15.html#example-rolls-independent",
|
||
"type": "Example",
|
||
"number": "33",
|
||
"title": "",
|
||
"body": " Continuing , recall and . We found that , so and are not independent. Now consider the event . We have: Therefore events are independent. "
|
||
},
|
||
{
|
||
"id": "notes-01-20",
|
||
"level": "1",
|
||
"url": "notes-01-20.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Tuesday, Jan 20",
|
||
"body": " Tuesday, Jan 20 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. HW 1 Q5 Write for the event that there's a fire and for the event that there's visible smoke. Then the information we're given can be interpreted as: In this case, we can use the simpler version of Bayes' Theorem: Unlike our usual diagnostic testing examples, we do have access to the denominator probability here. Sec 2.1: Random Variables A random variable is a function . The idea is that is a variable representing a real number value which depends on the outcome of an experiment. An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let be the height of plant . Let be the number of seeds that didn't sprout. Let A random variable has its own probability distribution. Roll a fair D6 twice. Let be the sum of the rolls. Then has 36 elements. Since the die is fair, the distribution on is uniform, i.e., for any . takes on the values , with probabilities: Distribution for 2 1\/36 3 2\/36 4 3\/36 7 6\/36 8 5\/36 12 1\/36 Note that the distribution on is uniform, but the distribution on is not. Let be a sample space and an event. Let is called an indicator random variable , and we say \" indicates \". The distribution on is: Suppose we flip a coin times. Let indicate heads on flip . Let be the total number of heads in all flips. Then: If is the probability of the coin coming up heads on a flip, then has the binomial distribution with parameters . We'll use the notation and: Suppose the coin has and we flip it times. Find . The relevant flip sequences are: Each individual sequence has a probability of . So the total probability is: That is, (number of flip sequences)(probability of each sequence). "
|
||
},
|
||
{
|
||
"id": "def-RV",
|
||
"level": "2",
|
||
"url": "notes-01-20.html#def-RV",
|
||
"type": "Definition",
|
||
"number": "34",
|
||
"title": "",
|
||
"body": " A random variable is a function . "
|
||
},
|
||
{
|
||
"id": "subsec-Discrete-Random-Variables-4",
|
||
"level": "2",
|
||
"url": "notes-01-20.html#subsec-Discrete-Random-Variables-4",
|
||
"type": "Example",
|
||
"number": "35",
|
||
"title": "",
|
||
"body": " An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let be the height of plant . Let be the number of seeds that didn't sprout. Let "
|
||
},
|
||
{
|
||
"id": "subsec-Discrete-Random-Variables-6",
|
||
"level": "2",
|
||
"url": "notes-01-20.html#subsec-Discrete-Random-Variables-6",
|
||
"type": "Example",
|
||
"number": "36",
|
||
"title": "",
|
||
"body": " Roll a fair D6 twice. Let be the sum of the rolls. Then has 36 elements. Since the die is fair, the distribution on is uniform, i.e., for any . takes on the values , with probabilities: Distribution for 2 1\/36 3 2\/36 4 3\/36 7 6\/36 8 5\/36 12 1\/36 Note that the distribution on is uniform, but the distribution on is not. "
|
||
},
|
||
{
|
||
"id": "subsec-Discrete-Random-Variables-7",
|
||
"level": "2",
|
||
"url": "notes-01-20.html#subsec-Discrete-Random-Variables-7",
|
||
"type": "Example",
|
||
"number": "38",
|
||
"title": "",
|
||
"body": " Let be a sample space and an event. Let is called an indicator random variable , and we say \" indicates \". The distribution on is: "
|
||
},
|
||
{
|
||
"id": "subsec-Discrete-Random-Variables-8",
|
||
"level": "2",
|
||
"url": "notes-01-20.html#subsec-Discrete-Random-Variables-8",
|
||
"type": "Example",
|
||
"number": "39",
|
||
"title": "",
|
||
"body": " Suppose we flip a coin times. Let indicate heads on flip . Let be the total number of heads in all flips. Then: If is the probability of the coin coming up heads on a flip, then has the binomial distribution with parameters . We'll use the notation and: Suppose the coin has and we flip it times. Find . The relevant flip sequences are: Each individual sequence has a probability of . So the total probability is: That is, (number of flip sequences)(probability of each sequence). "
|
||
},
|
||
{
|
||
"id": "notes-01-22",
|
||
"level": "1",
|
||
"url": "notes-01-22.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Thursday, Jan 22",
|
||
"body": " Thursday, Jan 22 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Discrete Distributions Suppose we expand . We'll get a polynomial with some coefficients: where , read \" choose \", is the coefficient of . Alternatively: is the number of ways to pick things out of a set of things. Then: where , read \" factorial\", refers to the product: So now we get a general probability formula for the binomial distribution: Let's see some more important distribution types: Suppose we have a coin with bias , i.e., probability of coming up heads. (Note: this is not a common term, but it will be convenient for us to have some terminology for it since we'll refer to this parameter often.) We flip the coin repeatedly until we see heads. Let be the number of flips. has the geometric distribution with parameter . We'll write , and we'll write: The geometric distribution is given by: Geometrid Distribution flip sequences 1 H 2 TH 3 TTH 4 TTTH In general: Suppose a toxin molecule inside a cell has a 0.2 chance of leaving during each minute. Let be the number of minutes until the molecule leaves. Find If , then is either 1, 2, or 3. A Poisson process is a process in which some event occurs at a constant probabilistic rate . Suppose we observe a Poisson process for time. Let be the number of occurrences of the event during that observation time. Then has the Poisson distribution with parameters . We'll write , and: Consider the following examples: A radioactive material emmitting particles as it decays. A call center receiving calls. A stretch of highways seeing cars pass by. Suppose a highway typically has 200 cars per hour. Let be the number of cars in a 30-minute observation period. Then , so . Then: Continuous Distributions Suppose we pick a real number uniformly. Intuitively, we can say things like and . What aobut Assigning probabilities to individual outcomes isn't useful here. Instead, we assign probabilities to intervals of the form . Let be a random variable. If takes an interval's worth of values, then it's called continuous . Otherwise, it's called discrete . Indicator, Bin, Geom, Poiss are all discrete. Pick is continuous. Height of a plant is continuous. Time is often (but not always) treated as continuous. Let be a continuous random variable. A probability density function (pdf) for is a function such that: for all , and, , where indicates that we should integrate over the entire range of possible values of . Given a pdf for : Pick uniformly. Because of the uniform assumption, must be a constant function for some . To find : More generally, if is chosen uniformly, then the pdf would be: "
|
||
},
|
||
{
|
||
"id": "thm-binomial",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#thm-binomial",
|
||
"type": "Theorem",
|
||
"number": "40",
|
||
"title": "",
|
||
"body": " Suppose we expand . We'll get a polynomial with some coefficients: where , read \" choose \", is the coefficient of . Alternatively: is the number of ways to pick things out of a set of things. Then: where , read \" factorial\", refers to the product: "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-3",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-3",
|
||
"type": "Fact",
|
||
"number": "41",
|
||
"title": "",
|
||
"body": " "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-4",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-4",
|
||
"type": "Example",
|
||
"number": "42",
|
||
"title": "",
|
||
"body": " "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-7",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-7",
|
||
"type": "Example",
|
||
"number": "43",
|
||
"title": "",
|
||
"body": " Suppose we have a coin with bias , i.e., probability of coming up heads. (Note: this is not a common term, but it will be convenient for us to have some terminology for it since we'll refer to this parameter often.) We flip the coin repeatedly until we see heads. Let be the number of flips. "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-8",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-8",
|
||
"type": "Definition",
|
||
"number": "44",
|
||
"title": "",
|
||
"body": " has the geometric distribution with parameter . We'll write , and we'll write: The geometric distribution is given by: Geometrid Distribution flip sequences 1 H 2 TH 3 TTH 4 TTTH In general: "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-9",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-9",
|
||
"type": "Example",
|
||
"number": "46",
|
||
"title": "",
|
||
"body": " Suppose a toxin molecule inside a cell has a 0.2 chance of leaving during each minute. Let be the number of minutes until the molecule leaves. Find If , then is either 1, 2, or 3. "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-10",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-10",
|
||
"type": "Definition",
|
||
"number": "47",
|
||
"title": "",
|
||
"body": " A Poisson process is a process in which some event occurs at a constant probabilistic rate . Suppose we observe a Poisson process for time. Let be the number of occurrences of the event during that observation time. Then has the Poisson distribution with parameters . We'll write , and: "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-11",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-11",
|
||
"type": "Example",
|
||
"number": "48",
|
||
"title": "",
|
||
"body": " Consider the following examples: A radioactive material emmitting particles as it decays. A call center receiving calls. A stretch of highways seeing cars pass by. "
|
||
},
|
||
{
|
||
"id": "subsec-discrete-distributions-12",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-discrete-distributions-12",
|
||
"type": "Example",
|
||
"number": "49",
|
||
"title": "",
|
||
"body": " Suppose a highway typically has 200 cars per hour. Let be the number of cars in a 30-minute observation period. Then , so . Then: "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-distributions-2",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-continuous-distributions-2",
|
||
"type": "Example",
|
||
"number": "50",
|
||
"title": "",
|
||
"body": " Suppose we pick a real number uniformly. Intuitively, we can say things like and . What aobut Assigning probabilities to individual outcomes isn't useful here. Instead, we assign probabilities to intervals of the form . "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-distributions-3",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-continuous-distributions-3",
|
||
"type": "Definition",
|
||
"number": "51",
|
||
"title": "",
|
||
"body": " Let be a random variable. If takes an interval's worth of values, then it's called continuous . Otherwise, it's called discrete . "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-distributions-4",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-continuous-distributions-4",
|
||
"type": "Example",
|
||
"number": "52",
|
||
"title": "",
|
||
"body": " Indicator, Bin, Geom, Poiss are all discrete. Pick is continuous. Height of a plant is continuous. Time is often (but not always) treated as continuous. "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-distributions-5",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-continuous-distributions-5",
|
||
"type": "Definition",
|
||
"number": "53",
|
||
"title": "",
|
||
"body": " Let be a continuous random variable. A probability density function (pdf) for is a function such that: for all , and, , where indicates that we should integrate over the entire range of possible values of . Given a pdf for : "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-distributions-6",
|
||
"level": "2",
|
||
"url": "notes-01-22.html#subsec-continuous-distributions-6",
|
||
"type": "Example",
|
||
"number": "54",
|
||
"title": "",
|
||
"body": " Pick uniformly. Because of the uniform assumption, must be a constant function for some . To find : More generally, if is chosen uniformly, then the pdf would be: "
|
||
},
|
||
{
|
||
"id": "notes-01-27",
|
||
"level": "1",
|
||
"url": "notes-01-27.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Tuesday, Jan 27",
|
||
"body": " Tuesday, Jan 27 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Continuous Distributions Let with . Find . Then, we can calculate probabilities, e.g.: Note: a pdf outputs probability densities, not probabilities. To get probabilities, we must integrate. Let be a continuous random variable with values in . The cumulative distribution function (cdf) is: To calculate it: Let . Then: Then, we can calculate probabilities, e.g.: Given , we can find by calculating: Given , we can find by calculating: Suppose a machine needs repairs on average twice per month. Let be the time until repair. This is a Poisson process ( per month). is said to have the exponential distribution with parameter . We'll write . The exponential density function is: Let . Then: So, e.g.: The distributions Bin, Geom, Poiss, and Exp are conceptually linked. Relationship of common distributions Counting Events Time Until Discrete Time Bin Geom Continuous Time Poiss Exp Joint Distributions Let take values and take values . The joint distribution of and is the collection of all values for every combination. The separarte distributions for and are called marginal distributions . Suppose , with joint distribution below. Example Joint Distribution 0.1 0.15 0.05 0.2 0.2 0.3 We find the marginal distribution for by summing along the columns: We find the marginal distribution for by summing along the rows: Random variables are independent if: for every combination. In the previous example: so are not independent. Let indicate heads on the first and second flip, respectively, of a fair coin. Then: Joint Distribution for Indicator Random Variables 1\/4 1\/4 1\/4 1\/4 Then the marginal distributions are: Then, for any : So are independent. "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-2",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-2",
|
||
"type": "Example",
|
||
"number": "55",
|
||
"title": "",
|
||
"body": " Let with . Find . Then, we can calculate probabilities, e.g.: "
|
||
},
|
||
{
|
||
"id": "def-cdf",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#def-cdf",
|
||
"type": "Definition",
|
||
"number": "56",
|
||
"title": "",
|
||
"body": " Let be a continuous random variable with values in . The cumulative distribution function (cdf) is: To calculate it: "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-5",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-5",
|
||
"type": "Example",
|
||
"number": "57",
|
||
"title": "",
|
||
"body": " Let . Then: Then, we can calculate probabilities, e.g.: "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-7",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-7",
|
||
"type": "Example",
|
||
"number": "58",
|
||
"title": "",
|
||
"body": " Suppose a machine needs repairs on average twice per month. Let be the time until repair. This is a Poisson process ( per month). "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-8",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-8",
|
||
"type": "Definition",
|
||
"number": "59",
|
||
"title": "",
|
||
"body": " is said to have the exponential distribution with parameter . We'll write . The exponential density function is: "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-9",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-9",
|
||
"type": "Example",
|
||
"number": "60",
|
||
"title": "",
|
||
"body": " Let . Then: So, e.g.: "
|
||
},
|
||
{
|
||
"id": "subsec-continuous-2-10",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-continuous-2-10",
|
||
"type": "Remark",
|
||
"number": "61",
|
||
"title": "",
|
||
"body": " The distributions Bin, Geom, Poiss, and Exp are conceptually linked. Relationship of common distributions Counting Events Time Until Discrete Time Bin Geom Continuous Time Poiss Exp "
|
||
},
|
||
{
|
||
"id": "subsec-Joint-Distributions-2",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-Joint-Distributions-2",
|
||
"type": "Definition",
|
||
"number": "63",
|
||
"title": "",
|
||
"body": " Let take values and take values . The joint distribution of and is the collection of all values for every combination. The separarte distributions for and are called marginal distributions . "
|
||
},
|
||
{
|
||
"id": "subsec-Joint-Distributions-3",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-Joint-Distributions-3",
|
||
"type": "Example",
|
||
"number": "64",
|
||
"title": "",
|
||
"body": " Suppose , with joint distribution below. Example Joint Distribution 0.1 0.15 0.05 0.2 0.2 0.3 We find the marginal distribution for by summing along the columns: We find the marginal distribution for by summing along the rows: "
|
||
},
|
||
{
|
||
"id": "subsec-Joint-Distributions-4",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-Joint-Distributions-4",
|
||
"type": "Definition",
|
||
"number": "66",
|
||
"title": "",
|
||
"body": " Random variables are independent if: for every combination. "
|
||
},
|
||
{
|
||
"id": "subsec-Joint-Distributions-5",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-Joint-Distributions-5",
|
||
"type": "Example",
|
||
"number": "67",
|
||
"title": "",
|
||
"body": " In the previous example: so are not independent. "
|
||
},
|
||
{
|
||
"id": "subsec-Joint-Distributions-6",
|
||
"level": "2",
|
||
"url": "notes-01-27.html#subsec-Joint-Distributions-6",
|
||
"type": "Example",
|
||
"number": "68",
|
||
"title": "",
|
||
"body": " Let indicate heads on the first and second flip, respectively, of a fair coin. Then: Joint Distribution for Indicator Random Variables 1\/4 1\/4 1\/4 1\/4 Then the marginal distributions are: Then, for any : So are independent. "
|
||
},
|
||
{
|
||
"id": "notes-01-29",
|
||
"level": "1",
|
||
"url": "notes-01-29.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Thursday, Jan 29",
|
||
"body": " Thursday, Jan 29 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Expected Value Let be the roll of a fair D6. What is the average value of ? The sample space is . To find the average: That is: take each value of , multiply by the probability, and add all the results together. Given a distribution table: Distribution 1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.5 Then the average is: Let be a discrete random variable taking values with probabilities . The expected value of is: Let indicate , with . Then: That is: \"E(indicator random variable) = Pr(event that it indicates)\". Suppose we flip a fair coin 100 times. Let indicate heads on each flip. Then , so for every . What if indicates a run of 4 heads starting at flip 3? That is, flips 3, 4, 5, and 6 must come up heads, and all other flips can come up either heads or tails. Since these flip results are independent, the probabilities multiply: Let be a continuous random variable with pdf . The expected value of is: It's worth putting this side-by-side with to compare the structure of each formula. These both say: \"multiply each value of the random variable by the probability, then accumulate all of those products\". Expected value is a weighted average of random variable values, with the probabilities as the weights. Let with pdf . Then: What would happen if you forgot the ? Then the calculation would become: This mistake will often be easy to catch. For example, this random variable takes values between 0 and 1, so it doesn't seem very likely that the average value is 1! Find given a distribution for . Distribution for -1 0.4 1 0.1 2 0.2 3 0.3 We can start by writing a distribution table for : Distribution for 1 0.5 4 0.2 9 0.3 Then: If takes on values with probabilities , and is any function, then: Revisiting the previous example, the theorem says we don't need to first create the distribution table for . We can use the distribution table for , and just apply the square to each value: The situation with continuous random variables is similar. Let with pdf . Then: Notice, in particular, that If are random variables with finite expected value and , then: Let be the number of heads in coin flips with bias . Then . Gross. Instead of calculating this directly, define to indicate heads on each flip. Then: Flip a fair coin 100 times. Then, the expected number of heads is: Flip a fair coin 100 times. What is the expected number of runs of 4 heads? As in the binomial EV calculation, define indicator random variables each indicating a run of 4 heads starting at the specified flip. Then for each . Let be the number of runs of 4 heads. Then: Consider a geometric distribution . Geometric Distribution with 1 2 3 4 Then is an infinite summation. Instead, consider the following argument. If we flip a coin until we see heads, we either see heads on flip 1 or not. In the first case, . In the second case, what is the average value of ? Starting at flip 2, it will take on average flips to see heads. Since we already flipped the coin once, the total number of flips will be . So, we can write: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-2",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-2",
|
||
"type": "Example",
|
||
"number": "70",
|
||
"title": "",
|
||
"body": " Let be the roll of a fair D6. What is the average value of ? The sample space is . To find the average: That is: take each value of , multiply by the probability, and add all the results together. "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-3",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-3",
|
||
"type": "Example",
|
||
"number": "71",
|
||
"title": "",
|
||
"body": " Given a distribution table: Distribution 1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.5 Then the average is: "
|
||
},
|
||
{
|
||
"id": "def-discrete-EV",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#def-discrete-EV",
|
||
"type": "Definition",
|
||
"number": "73",
|
||
"title": "",
|
||
"body": " Let be a discrete random variable taking values with probabilities . The expected value of is: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-5",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-5",
|
||
"type": "Example",
|
||
"number": "74",
|
||
"title": "",
|
||
"body": " Let indicate , with . Then: That is: \"E(indicator random variable) = Pr(event that it indicates)\". "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-6",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-6",
|
||
"type": "Example",
|
||
"number": "75",
|
||
"title": "",
|
||
"body": " Suppose we flip a fair coin 100 times. Let indicate heads on each flip. Then , so for every . What if indicates a run of 4 heads starting at flip 3? That is, flips 3, 4, 5, and 6 must come up heads, and all other flips can come up either heads or tails. Since these flip results are independent, the probabilities multiply: "
|
||
},
|
||
{
|
||
"id": "def-continuous-EV",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#def-continuous-EV",
|
||
"type": "Definition",
|
||
"number": "76",
|
||
"title": "",
|
||
"body": " Let be a continuous random variable with pdf . The expected value of is: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-9",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-9",
|
||
"type": "Example",
|
||
"number": "77",
|
||
"title": "",
|
||
"body": " Let with pdf . Then: What would happen if you forgot the ? Then the calculation would become: This mistake will often be easy to catch. For example, this random variable takes values between 0 and 1, so it doesn't seem very likely that the average value is 1! "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-10",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-10",
|
||
"type": "Example",
|
||
"number": "78",
|
||
"title": "",
|
||
"body": " Find given a distribution for . Distribution for -1 0.4 1 0.1 2 0.2 3 0.3 We can start by writing a distribution table for : Distribution for 1 0.5 4 0.2 9 0.3 Then: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-11",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-11",
|
||
"type": "Theorem",
|
||
"number": "81",
|
||
"title": "",
|
||
"body": " If takes on values with probabilities , and is any function, then: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-12",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-12",
|
||
"type": "Example",
|
||
"number": "82",
|
||
"title": "",
|
||
"body": " Revisiting the previous example, the theorem says we don't need to first create the distribution table for . We can use the distribution table for , and just apply the square to each value: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-13",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-13",
|
||
"type": "Example",
|
||
"number": "83",
|
||
"title": "",
|
||
"body": " The situation with continuous random variables is similar. Let with pdf . Then: Notice, in particular, that "
|
||
},
|
||
{
|
||
"id": "thm-linearity",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#thm-linearity",
|
||
"type": "Theorem",
|
||
"number": "84",
|
||
"title": "",
|
||
"body": " If are random variables with finite expected value and , then: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-15",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-15",
|
||
"type": "Example",
|
||
"number": "85",
|
||
"title": "",
|
||
"body": " Let be the number of heads in coin flips with bias . Then . Gross. Instead of calculating this directly, define to indicate heads on each flip. Then: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-16",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-16",
|
||
"type": "Example",
|
||
"number": "86",
|
||
"title": "",
|
||
"body": " Flip a fair coin 100 times. Then, the expected number of heads is: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-17",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-17",
|
||
"type": "Example",
|
||
"number": "87",
|
||
"title": "",
|
||
"body": " Flip a fair coin 100 times. What is the expected number of runs of 4 heads? As in the binomial EV calculation, define indicator random variables each indicating a run of 4 heads starting at the specified flip. Then for each . Let be the number of runs of 4 heads. Then: "
|
||
},
|
||
{
|
||
"id": "notes-01-29-3-18",
|
||
"level": "2",
|
||
"url": "notes-01-29.html#notes-01-29-3-18",
|
||
"type": "Example",
|
||
"number": "88",
|
||
"title": "",
|
||
"body": " Consider a geometric distribution . Geometric Distribution with 1 2 3 4 Then is an infinite summation. Instead, consider the following argument. If we flip a coin until we see heads, we either see heads on flip 1 or not. In the first case, . In the second case, what is the average value of ? Starting at flip 2, it will take on average flips to see heads. Since we already flipped the coin once, the total number of flips will be . So, we can write: "
|
||
},
|
||
{
|
||
"id": "notes-02-03",
|
||
"level": "1",
|
||
"url": "notes-02-03.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Tuesday, Feb 3",
|
||
"body": " Tuesday, Feb 3 This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes. Variance Question: How spread out are values? One answer we might try is to measure the average distance from the average value: To simplify notation, we'll write . Also, it's often usefull to square a term rather than take absolute value when we want to ensure a positive output, so we'll define... Let be a random variable with . The variance of is: is called the standard deviation . "
|
||
},
|
||
{
|
||
"id": "def-variance",
|
||
"level": "2",
|
||
"url": "notes-02-03.html#def-variance",
|
||
"type": "Definition",
|
||
"number": "90",
|
||
"title": "",
|
||
"body": " Let be a random variable with . The variance of is: is called the standard deviation . "
|
||
},
|
||
{
|
||
"id": "quiz-01",
|
||
"level": "1",
|
||
"url": "quiz-01.html",
|
||
"type": "Worksheet",
|
||
"number": "",
|
||
"title": "Quiz 1",
|
||
"body": " Quiz 1 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. If and are any sets, then and . True. If and are any events, then . False. If and are any events, then . False. We find a 4-sided die with faces 0, 1, 3, and 5. An experiment consists of rolling the die two times. Write down the sample space of all possible outcomes for this experiment. Let be the event that the second roll is at least twice the value of the first roll. Let m B be the event that the sum of the rolls is odd. Assume the die is fair. List the outcomes in and calculate . List the outcomes in and calculate . Then , and . A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time. For a patient who receives a positive test, what is the probability they have the disease? Let be the event of receiving a positive test result and be the event of having the disease. The given information is: , and . Then, using Bayes' Theorem: "
|
||
},
|
||
{
|
||
"id": "quiz-01-3-1",
|
||
"level": "2",
|
||
"url": "quiz-01.html#quiz-01-3-1",
|
||
"type": "Worksheet Exercise",
|
||
"number": "1",
|
||
"title": "",
|
||
"body": " Write either True or False for each of the following statements. No justification is required. If and are any sets, then and . True. If and are any events, then . False. If and are any events, then . False. "
|
||
},
|
||
{
|
||
"id": "quiz-01-3-2",
|
||
"level": "2",
|
||
"url": "quiz-01.html#quiz-01-3-2",
|
||
"type": "Worksheet Exercise",
|
||
"number": "2",
|
||
"title": "",
|
||
"body": " We find a 4-sided die with faces 0, 1, 3, and 5. An experiment consists of rolling the die two times. Write down the sample space of all possible outcomes for this experiment. Let be the event that the second roll is at least twice the value of the first roll. Let m B be the event that the sum of the rolls is odd. Assume the die is fair. List the outcomes in and calculate . List the outcomes in and calculate . Then , and . "
|
||
},
|
||
{
|
||
"id": "quiz-01-4-1",
|
||
"level": "2",
|
||
"url": "quiz-01.html#quiz-01-4-1",
|
||
"type": "Worksheet Exercise",
|
||
"number": "3",
|
||
"title": "",
|
||
"body": " A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time. For a patient who receives a positive test, what is the probability they have the disease? Let be the event of receiving a positive test result and be the event of having the disease. The given information is: , and . Then, using Bayes' Theorem: "
|
||
},
|
||
{
|
||
"id": "quiz-02",
|
||
"level": "1",
|
||
"url": "quiz-02.html",
|
||
"type": "Worksheet",
|
||
"number": "",
|
||
"title": "Quiz 2",
|
||
"body": " Quiz 2 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Checking each cell in the table: So are independent. Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . "
|
||
},
|
||
{
|
||
"id": "quiz-02-3-1",
|
||
"level": "2",
|
||
"url": "quiz-02.html#quiz-02-3-1",
|
||
"type": "Worksheet Exercise",
|
||
"number": "1",
|
||
"title": "",
|
||
"body": " Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. "
|
||
},
|
||
{
|
||
"id": "quiz-02-3-2",
|
||
"level": "2",
|
||
"url": "quiz-02.html#quiz-02-3-2",
|
||
"type": "Worksheet Exercise",
|
||
"number": "2",
|
||
"title": "",
|
||
"body": " Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Checking each cell in the table: So are independent. "
|
||
},
|
||
{
|
||
"id": "quiz-02-4-1",
|
||
"level": "2",
|
||
"url": "quiz-02.html#quiz-02-4-1",
|
||
"type": "Worksheet Exercise",
|
||
"number": "3",
|
||
"title": "",
|
||
"body": " Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review",
|
||
"level": "1",
|
||
"url": "recitation-calculus-review.html",
|
||
"type": "Worksheet",
|
||
"number": "",
|
||
"title": "Calculus Review",
|
||
"body": " Calculus Review In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations. The Power Rule for derivatives states: The Power Rule for antiderivatives states: Practice with the following derivatives: Practice with the following antiderivatives: Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: If is a probability density function (don't worry about what that means for now), then integrals such as are used to calculate certain probabilities, as are values of the form . Practice with the following integrals: "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review-3-1",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#recitation-calculus-review-3-1",
|
||
"type": "Worksheet Exercise",
|
||
"number": "1",
|
||
"title": "",
|
||
"body": " The Power Rule for derivatives states: "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review-3-2",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#recitation-calculus-review-3-2",
|
||
"type": "Worksheet Exercise",
|
||
"number": "2",
|
||
"title": "",
|
||
"body": " The Power Rule for antiderivatives states: "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review-3-3",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#recitation-calculus-review-3-3",
|
||
"type": "Worksheet Exercise",
|
||
"number": "3",
|
||
"title": "",
|
||
"body": " Practice with the following derivatives: "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review-3-4",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#recitation-calculus-review-3-4",
|
||
"type": "Worksheet Exercise",
|
||
"number": "4",
|
||
"title": "",
|
||
"body": " Practice with the following antiderivatives: "
|
||
},
|
||
{
|
||
"id": "thm-FTC",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#thm-FTC",
|
||
"type": "Theorem",
|
||
"number": "92",
|
||
"title": "Fundamental Theorem of Calculus.",
|
||
"body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: "
|
||
},
|
||
{
|
||
"id": "recitation-calculus-review-4-3",
|
||
"level": "2",
|
||
"url": "recitation-calculus-review.html#recitation-calculus-review-4-3",
|
||
"type": "Worksheet Exercise",
|
||
"number": "5",
|
||
"title": "",
|
||
"body": " Practice with the following integrals: "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review",
|
||
"level": "1",
|
||
"url": "Exam-1-Review.html",
|
||
"type": "Section",
|
||
"number": "",
|
||
"title": "Exam 1 Review",
|
||
"body": " Exam 1 Review Use the following problems to prepare for the exam. There will be in-class review on Thursday, February 12. Your recitation this week will also be exam review. Allowed Materials You will be allowed to use a scientific calculator ( not a graphing calculator, not a calculator app on your phone). You may not share a calculator with another student; you must use your own calculator. You may bring a standard 3 in x 5 in index card with prepared notes. You may use both sides of the notecard. You must put your full name in the top right corner of the card, and turn it in along with your exam. Consider the sets , , and , which are all subsets of . Find . Find , , , , , , , and . Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? , , , , , , , . In particular, note that , so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets. Find and . , . Suppose we have a 6-sided die that's weighted to roll a 6 half of the time. We roll the die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result ---rolling a 4 and then a 2.] Note that simply lists outcomes with no reference to the probabilities. So the answer here is the same as in . Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be? For two flips: . For three flips: . For four flips: Each additional flip doubles the number of outcomes. So, with ten flips, we'll have . If we roll a 6-sided die ten times, how many possible results will there be? Each additional roll will multiply the number of outcomes by 6. So, with 10 rolls, we'll have Consider the sample space with probability distribution below. Calculate the probabilities of , , , and . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 Suppose a die has the values on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die. Probability Distribution for a Linearly Scaled Die 1 2 3 4 5 6 Suppose a die has the values on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die. Probability Distribution for an Even-biased Die 1 2 3 4 5 6 A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of , find the probability of the toxin molecule leaving the cell during the th minute. What is the probability of the molecule leaving the cell during the first 3 minutes? For short, write to mean the probability of the toxin molecule leaving during the th minute. Then The probability of leaving during the first 3 minutes is In each of the following scenarios with given events and , alculate , , , and . An experiment consists of rolling a fair die two times. Let be the event that the sum is even, and let be the event that the second roll is higher than the first. So , , and . Finally: An experiment consists of flipping a fair coin three times. Let be the event that the first and second flips match. Let be the event that there are at least two heads. So , , and . Finally: A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time. For a patient who receives a positive test, what is the probability they have the disease? Let be the event of testing positive and the event of having the disease. Then the prevalence is given as 3.2%, or 0.032. The sensitivity is , and the specificity is . So, according to Bayes' Theorem: For a patient who receives a negative test, what is the probability they do not have the disease? An experiment consists of rolling a fair die two times. Let be the event that the sum is even, and let be the event that the second roll is higher than the first. Are and independent? So , , and . Finally: so and are not independent. An experiment consists of flipping a fair coin three times. Let be the event that the first and second flips match. Let be the event that there are at least two heads. Are and independent? So , , and . Finally: so and are independent. Let and be events in the sample space . Create a probability distribution for so that are independent. Example Distribution 1 0.1 2 0.2 3 0.2 4 0.1 5 0.1 6 0.3 Now , , and so and are independent. An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability , find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads. Let be the number of heads. Then , so: An experiment consists of flipping a coin repeatedly until we first see heads. If the coin comes up heads with probability 0.4, what is the probability we'll see our first heads within three flips? What about precisely on the third flip? Let be the number of flips until we see heads. Then is geometric with parameter , so: Which flip has the highest chance of being the first flip to come up heads? A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers. A continuous random variable taking values in has p.d.f. for some constant . What is the value of ? Find A continuous random variable taking values in has p.d.f. . Find the c.d.f. . Use your c.d.f. to find . . A continuous random variable taking values in has c.d.f. . Find the p.d.f. . Consider with the joint distribution table below. Are independent? Joint distribution for 0.2 0.3 0.4 0.1 No. For example, Suppose have the distributions: Assuming are independent, write a joint distribution table. Joint Distribution 0.04 0.16 0.2 0.06 0.24 0.3 Consider a random variable with probability distribution below. Find . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 4.8. Suppose we flip a coin times, and let count the number of heads. If the coin comes up heads on a flip with probability , what is ? 40. What if and ? 48. What if and ? 100. If , , and , what is ? 4. A continuous random variable taking values in has p.d.f. . What is ? 2\/3. A continuous random variable taking values in has p.d.f. . What is ? A continuous random variable taking values in has p.d.f. . Find . Consider a random variable with probability distribution below. Find . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 Suppose we flip a coin times, and let count the number of heads. If the coin comes up heads on a flip with probability , what is ? What if and ? What if and ? If , , what is ? A continuous random variable taking values in has p.d.f. . What is ? A continuous random variable taking values in has p.d.f. . What is ? A continuous random variable taking values in has p.d.f. . Find . Suppose a coin has an unknown probability of coming up heads. We perform the experiment in independent trials, during which it takes flips to see our first heads in each trial. Find a \"common sense\" MLE formula for the geometric distribution. A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate? A radioactive material emits particles at an unknown probabilistic rate particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function based on this data. What is the maximum likelihood estimation for ? Suppose a parameter takes values in with likelihood function . Find the maximum likelihood estimation of . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-1",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-1",
|
||
"type": "Exercise",
|
||
"number": "1",
|
||
"title": "",
|
||
"body": " Consider the sets , , and , which are all subsets of . Find . Find , , , , , , , and . Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? , , , , , , , . In particular, note that , so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets. Find and . , . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-2",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-2",
|
||
"type": "Exercise",
|
||
"number": "2",
|
||
"title": "",
|
||
"body": " Suppose we have a 6-sided die that's weighted to roll a 6 half of the time. We roll the die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result ---rolling a 4 and then a 2.] Note that simply lists outcomes with no reference to the probabilities. So the answer here is the same as in . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-3",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-3",
|
||
"type": "Exercise",
|
||
"number": "3",
|
||
"title": "",
|
||
"body": " Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be? For two flips: . For three flips: . For four flips: Each additional flip doubles the number of outcomes. So, with ten flips, we'll have . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-4",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-4",
|
||
"type": "Exercise",
|
||
"number": "4",
|
||
"title": "",
|
||
"body": " If we roll a 6-sided die ten times, how many possible results will there be? Each additional roll will multiply the number of outcomes by 6. So, with 10 rolls, we'll have "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-5",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-5",
|
||
"type": "Exercise",
|
||
"number": "5",
|
||
"title": "",
|
||
"body": " Consider the sample space with probability distribution below. Calculate the probabilities of , , , and . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-6",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-6",
|
||
"type": "Exercise",
|
||
"number": "6",
|
||
"title": "",
|
||
"body": " Suppose a die has the values on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die. Probability Distribution for a Linearly Scaled Die 1 2 3 4 5 6 "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-7",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-7",
|
||
"type": "Exercise",
|
||
"number": "7",
|
||
"title": "",
|
||
"body": " Suppose a die has the values on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die. Probability Distribution for an Even-biased Die 1 2 3 4 5 6 "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-8",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-8",
|
||
"type": "Exercise",
|
||
"number": "8",
|
||
"title": "",
|
||
"body": " A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of , find the probability of the toxin molecule leaving the cell during the th minute. What is the probability of the molecule leaving the cell during the first 3 minutes? For short, write to mean the probability of the toxin molecule leaving during the th minute. Then The probability of leaving during the first 3 minutes is "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-9-2",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-9-2",
|
||
"type": "Exercise",
|
||
"number": "9",
|
||
"title": "",
|
||
"body": " An experiment consists of rolling a fair die two times. Let be the event that the sum is even, and let be the event that the second roll is higher than the first. So , , and . Finally: "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-9-3",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-9-3",
|
||
"type": "Exercise",
|
||
"number": "10",
|
||
"title": "",
|
||
"body": " An experiment consists of flipping a fair coin three times. Let be the event that the first and second flips match. Let be the event that there are at least two heads. So , , and . Finally: "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-10",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-10",
|
||
"type": "Exercise",
|
||
"number": "11",
|
||
"title": "",
|
||
"body": " A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time. For a patient who receives a positive test, what is the probability they have the disease? Let be the event of testing positive and the event of having the disease. Then the prevalence is given as 3.2%, or 0.032. The sensitivity is , and the specificity is . So, according to Bayes' Theorem: For a patient who receives a negative test, what is the probability they do not have the disease? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-11",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-11",
|
||
"type": "Exercise",
|
||
"number": "12",
|
||
"title": "",
|
||
"body": " An experiment consists of rolling a fair die two times. Let be the event that the sum is even, and let be the event that the second roll is higher than the first. Are and independent? So , , and . Finally: so and are not independent. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-12",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-12",
|
||
"type": "Exercise",
|
||
"number": "13",
|
||
"title": "",
|
||
"body": " An experiment consists of flipping a fair coin three times. Let be the event that the first and second flips match. Let be the event that there are at least two heads. Are and independent? So , , and . Finally: so and are independent. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-13",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-13",
|
||
"type": "Exercise",
|
||
"number": "14",
|
||
"title": "",
|
||
"body": " Let and be events in the sample space . Create a probability distribution for so that are independent. Example Distribution 1 0.1 2 0.2 3 0.2 4 0.1 5 0.1 6 0.3 Now , , and so and are independent. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-14",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-14",
|
||
"type": "Exercise",
|
||
"number": "15",
|
||
"title": "",
|
||
"body": " An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability , find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads. Let be the number of heads. Then , so: "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-15",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-15",
|
||
"type": "Exercise",
|
||
"number": "16",
|
||
"title": "",
|
||
"body": " An experiment consists of flipping a coin repeatedly until we first see heads. If the coin comes up heads with probability 0.4, what is the probability we'll see our first heads within three flips? What about precisely on the third flip? Let be the number of flips until we see heads. Then is geometric with parameter , so: Which flip has the highest chance of being the first flip to come up heads? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-16",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-16",
|
||
"type": "Exercise",
|
||
"number": "17",
|
||
"title": "",
|
||
"body": " A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-17",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-17",
|
||
"type": "Exercise",
|
||
"number": "18",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. for some constant . What is the value of ? Find "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-18",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-18",
|
||
"type": "Exercise",
|
||
"number": "19",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . Find the c.d.f. . Use your c.d.f. to find . . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-19",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-19",
|
||
"type": "Exercise",
|
||
"number": "20",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has c.d.f. . Find the p.d.f. . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-20",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-20",
|
||
"type": "Exercise",
|
||
"number": "21",
|
||
"title": "",
|
||
"body": " Consider with the joint distribution table below. Are independent? Joint distribution for 0.2 0.3 0.4 0.1 No. For example, "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-21",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-21",
|
||
"type": "Exercise",
|
||
"number": "22",
|
||
"title": "",
|
||
"body": " Suppose have the distributions: Assuming are independent, write a joint distribution table. Joint Distribution 0.04 0.16 0.2 0.06 0.24 0.3 "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-22",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-22",
|
||
"type": "Exercise",
|
||
"number": "23",
|
||
"title": "",
|
||
"body": " Consider a random variable with probability distribution below. Find . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 4.8. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-23",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-23",
|
||
"type": "Exercise",
|
||
"number": "24",
|
||
"title": "",
|
||
"body": " Suppose we flip a coin times, and let count the number of heads. If the coin comes up heads on a flip with probability , what is ? 40. What if and ? 48. What if and ? 100. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-24",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-24",
|
||
"type": "Exercise",
|
||
"number": "25",
|
||
"title": "",
|
||
"body": " If , , and , what is ? 4. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-25",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-25",
|
||
"type": "Exercise",
|
||
"number": "26",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . What is ? 2\/3. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-26",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-26",
|
||
"type": "Exercise",
|
||
"number": "27",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . What is ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-27",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-27",
|
||
"type": "Exercise",
|
||
"number": "28",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . Find . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-28",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-28",
|
||
"type": "Exercise",
|
||
"number": "29",
|
||
"title": "",
|
||
"body": " Consider a random variable with probability distribution below. Find . 1 0.1 2 0.05 3 0.2 4 0.15 5 0.15 6 0.1 7 0.05 8 0.1 9 0.1 "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-29",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-29",
|
||
"type": "Exercise",
|
||
"number": "30",
|
||
"title": "",
|
||
"body": " Suppose we flip a coin times, and let count the number of heads. If the coin comes up heads on a flip with probability , what is ? What if and ? What if and ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-30",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-30",
|
||
"type": "Exercise",
|
||
"number": "31",
|
||
"title": "",
|
||
"body": " If , , what is ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-31",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-31",
|
||
"type": "Exercise",
|
||
"number": "32",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . What is ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-32",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-32",
|
||
"type": "Exercise",
|
||
"number": "33",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . What is ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-33",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-33",
|
||
"type": "Exercise",
|
||
"number": "34",
|
||
"title": "",
|
||
"body": " A continuous random variable taking values in has p.d.f. . Find . "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-34",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-34",
|
||
"type": "Exercise",
|
||
"number": "35",
|
||
"title": "",
|
||
"body": " Suppose a coin has an unknown probability of coming up heads. We perform the experiment in independent trials, during which it takes flips to see our first heads in each trial. Find a \"common sense\" MLE formula for the geometric distribution. "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-35",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-35",
|
||
"type": "Exercise",
|
||
"number": "36",
|
||
"title": "",
|
||
"body": " A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-36",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-36",
|
||
"type": "Exercise",
|
||
"number": "37",
|
||
"title": "",
|
||
"body": " A radioactive material emits particles at an unknown probabilistic rate particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function based on this data. What is the maximum likelihood estimation for ? "
|
||
},
|
||
{
|
||
"id": "Exam-1-Review-4-37",
|
||
"level": "2",
|
||
"url": "Exam-1-Review.html#Exam-1-Review-4-37",
|
||
"type": "Exercise",
|
||
"number": "38",
|
||
"title": "",
|
||
"body": " Suppose a parameter takes values in with likelihood function . Find the maximum likelihood estimation of . "
|
||
}
|
||
]
|
||
|
||
var ptx_lunr_idx = lunr(function () {
|
||
this.ref('id')
|
||
this.field('title')
|
||
this.field('body')
|
||
this.metadataWhitelist = ['position']
|
||
|
||
ptx_lunr_docs.forEach(function (doc) {
|
||
this.add(doc)
|
||
}, this)
|
||
})
|