diff --git a/publication/publication.ptx b/publication/publication.ptx index 9bc0fbe..980bb88 100644 --- a/publication/publication.ptx +++ b/publication/publication.ptx @@ -120,11 +120,11 @@ - + /> diff --git a/source/main.ptx b/source/main.ptx index 66d2ff0..8a04378 100644 --- a/source/main.ptx +++ b/source/main.ptx @@ -19,6 +19,12 @@ + + Class Notes + + + + Recitations diff --git a/source/notes/week01.ptx b/source/notes/week01.ptx index 8fda3ba..6f7e5f3 100644 --- a/source/notes/week01.ptx +++ b/source/notes/week01.ptx @@ -6,30 +6,303 @@

This is an outline of the topics we covered in the first week of 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.

- + - Monday 8/22 + Tuesday 1/13 + +

+ In calculus, you mostly asked deterministic questions about functions. + In science experiments, you need to take randomness into account. +

+ + + +

+ 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. +

+ + + + + +

+ Based on historical year-over-year population growth rate data for a particular species, what is a reasonable range for next year's population? +

+ + + + + +

+ The sample space, often denoted \Omega, 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. +

+ + + + + +

+ Let A be a set. + The symbol \in means "is an element of", as in a \in A. + Given another set B, we say A is a subset of B, written A\subset B, to mean that every element of the set A is also an element of the set B. +

+ + + + + +

+ An experiment consists of rolling a standard 6-sided die. + The sample space is \Omega = \{1, 2, 3, 4, 5, 6\}. + One possible event is A = \{2, 4, 6\}, i.e., the event that the result of the roll is even. +

+ + + + + +

+ Consider sets A and B, each contained inside \Omega. + We can combine sets in a variety of ways:

+
  • + Union + +

    + The union of A and B is the set A \cup B = \{x \mid x \in A \text{ or } x \in B\}. +

    +
    • + +
  • + Intersection + +

    + The intersection of A and B is the set A \cap B = \{x \mid x \in A \text{ and } x \in B\}. +

    +
    • + +
  • + Difference + +

    + The set difference A-B is the set A - B = \{x \mid x \in A \text{ and } x \notin B\}. +

    +
    • + +
  • + Complement + +

    + The complement of A is the set A^c = \{x \in \Omega \mid x \notin A\}. +

    +
    • + +
  • + Empty Set + +

    + The empty set, usually written \emptyset or \{\}, is the set which contains no elements. +

    +
    • +
    +

    +     +     + +

    + It's useful sometimes to draw pictures called Venn diagrams representing sets: +

    + +
    + Example Venn Diagram + + +

    + Venn diagram showing sets A, B, C with the region representing (A\cup B\cup C) - (A \cap C) shaded. +

    +     + + \begin{tikzpicture} + \def\firstcircle{(90:1.75cm) circle (2.5cm)} + \def\secondcircle{(210:1.75cm) circle (2.5cm)} + \def\thirdcircle{(330:1.75cm) circle (2.5cm)} + \fill[gray!30] \firstcircle; + \fill[gray!30] \secondcircle; + \fill[gray!30] \thirdcircle; + \begin{scope} + \clip \firstcircle; + \clip \thirdcircle; + \fill[white] \firstcircle; + \end{scope} + \draw \firstcircle node[text=black,above] {$A$}; + \draw \secondcircle node [text=black,below left] {$B$}; + \draw \thirdcircle node [text=black,below right] {$C$}; + \node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$}; + \end{tikzpicture} + + +
    + +

    + 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 standard 6-sided die. + The sample space is \Omega = \{1, 2, 3, 4, 5, 6\}. + We might assign probabilities as follows: +

    + + + Distribution for a fair die + + + + x + \Pr(x) + + + + 1 + 1/6 + + + + 2 + 1/6 + + + + 3 + 1/6 + + + + 4 + 1/6 + + + + 5 + 1/6 + + + + 6 + 1/6 + + +
    + +

    + 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 A = \{2, 4, 6\}, then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event: + + \Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} + + Note that this is the same result that we would get if we counted the total number of outcomes in A and divided by the total number of outcomes in \Omega. +

    +     +     + + + +

    + Given a finite set A, the cardinality of A, written |A|, is the number of elements in A. +

    +     +     + + + +

    + If \Omega is a finite probability space with the uniform distribution and A \subset \Omega is an event, then: + + \Pr(A) = \frac{|A|}{|\Omega|} + +

    +     +     + + + +

    + Suppose we have a weighted die that's much more likely to come up 6 than any other outcome. +

    + + + Distribution for a fair die + + + + x + \Pr(x) + + + + 1 + 0.1 + + + + 2 + 0.1 + + + + 3 + 0.1 + + + + 4 + 0.2 + + + + 5 + 0.1 + + + + 6 + 0.4 + + +
    + +

    + With this distribution, if A = \{2, 4, 6\}, then: + + \Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}. + +

    +     +     +     + + \ No newline at end of file