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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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@@ -231,6 +235,7 @@ eBookConfig.enable_chatcodes = false;
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<li><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></li>
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<li><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></li>
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<li><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></li>
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<li><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></li>
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</ul></nav><div class="autopermalink" data-description="Chapter: Class Notes"><a href="#ch-notes" title="Copy heading and permalink for Chapter: Class Notes" aria-label="Copy heading and permalink for Chapter: Class Notes">🔗</a></div></section></div>
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<div id="ptx-content-footer" class="ptx-content-footer">
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<a class="previous-button button" href="syllabus.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="notes-01-13.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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</li>
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<li class="toc-item toc-chapter">
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@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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</ul>
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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</li>
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">73</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para logical">
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<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> with probabilities <span class="process-math">\(p_1, p_2, \dotsc, p_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> of <span class="process-math">\(X\)</span> is:</div>
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<div class="displaymath process-math">
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\begin{gather*}
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\E(X) = \sum_{i=1}^n x_ip_i = x_1p_1 + x_2p_2 + \dotsb + x_np_n.
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\end{gather*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Definition 73"><a href="#def-discrete-EV" title="Copy heading and permalink for Definition 73" aria-label="Copy heading and permalink for Definition 73">🔗</a></div></article><span class="incontext"><a class="internal" href="notes-01-29.html#def-discrete-EV">in-context</a></span>
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"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. "
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"title": "Thursday, Jan 29",
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"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",
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||||
"level": "2",
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||||
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"type": "Example",
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||||
"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",
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||||
"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: "
|
||||
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|
||||
{
|
||||
"id": "def-discrete-EV",
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"level": "2",
|
||||
"url": "notes-01-29.html#def-discrete-EV",
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||||
"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",
|
||||
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|
||||
"body": " If takes on values with probabilities , and is any function, then: "
|
||||
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|
||||
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||||
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|
||||
"level": "2",
|
||||
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|
||||
"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",
|
||||
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|
||||
"body": " The situation with continuous random variables is similar. Let with pdf . Then: Notice, in particular, that "
|
||||
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|
||||
{
|
||||
"id": "thm-linearity",
|
||||
"level": "2",
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||||
"url": "notes-01-29.html#thm-linearity",
|
||||
"type": "Theorem",
|
||||
"number": "84",
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||||
"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": "quiz-01",
|
||||
"level": "1",
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||||
@@ -698,7 +851,7 @@ var ptx_lunr_docs = [
|
||||
"level": "2",
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||||
"url": "recitation-calculus-review.html#thm-FTC",
|
||||
"type": "Theorem",
|
||||
"number": "70",
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||||
"number": "90",
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||||
"title": "Fundamental Theorem of Calculus.",
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||||
"body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: "
|
||||
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|
||||
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||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
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||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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</li>
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||||
</ul>
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||||
</li>
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<li class="toc-item toc-chapter">
|
||||
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@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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||||
</li>
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||||
</ul>
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</li>
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||||
<li class="toc-item toc-chapter">
|
||||
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||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
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||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
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|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
|
||||
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|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
|
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|
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|
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-hw1-review" class="internal"><span class="title">HW 1 Q5</span></a></div></li>
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||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li>
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|
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<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
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<ul class="structural toc-item-list">
|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list active"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul></nav></div>
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-29"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Thursday, Jan 29</span>
|
||||
</h2>
|
||||
<section class="introduction" id="notes-01-29-2"><div class="para" id="notes-01-29-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> 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.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></section><section class="subsection" id="notes-01-29-3"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Expected Value</span>
|
||||
</h3>
|
||||
<article class="example example-like" id="notes-01-29-3-2"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">70</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-2-1-1">
|
||||
<div class="para">Let <span class="process-math">\(R\)</span> be the roll of a fair D6. What is the average value of <span class="process-math">\(R\text{?}\)</span> The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> To find the average:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-2-1-1-4">
|
||||
\begin{align*}
|
||||
\text{Avg} \amp = 1\left(\frac{1}{6}\right) + 2\left(\frac{1}{6}\right) + 3\left(\frac{1}{6}\right) + 4\left(\frac{1}{6}\right) + 5\left(\frac{1}{6}\right) + 6\left(\frac{1}{6}\right) \\
|
||||
\amp = \frac{21}{6} = \frac{7}{2} = 3.5
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">That is: take each value of <span class="process-math">\(R\text{,}\)</span> multiply by the probability, and add all the results together.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 70"><a href="#notes-01-29-3-2" title="Copy heading and permalink for Example 70" aria-label="Copy heading and permalink for Example 70">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-3"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">71</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-01-29-3-3-1-1">Given a distribution table:<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-01-29-3-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">72<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution<div class="autopermalink" data-description="Table 72: Distribution"><a href="#notes-01-29-3-3-1-2" title="Copy heading and permalink for Table 72: Distribution" aria-label="Copy heading and permalink for Table 72: Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(s\)</span></td>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = s)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">6</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.5</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="notes-01-29-3-3-1-3">
|
||||
<div class="para">Then the average is:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-3-1-3-1">
|
||||
\begin{gather*}
|
||||
\text{Avg} = 1(0.1) + 2(0.1) + 3(0.1) + 4(0.1) + 5(0.1) + 6(0.5) = 4.5.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 71"><a href="#notes-01-29-3-3" title="Copy heading and permalink for Example 71" aria-label="Copy heading and permalink for Example 71">🔗</a></div></article><article class="definition definition-like" id="def-discrete-EV"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">73</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-discrete-EV-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a discrete random variable taking values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> with probabilities <span class="process-math">\(p_1, p_2, \dotsc, p_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> of <span class="process-math">\(X\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="def-discrete-EV-1-1-6">
|
||||
\begin{gather*}
|
||||
\E(X) = \sum_{i=1}^n x_ip_i = x_1p_1 + x_2p_2 + \dotsb + x_np_n.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 73"><a href="#def-discrete-EV" title="Copy heading and permalink for Definition 73" aria-label="Copy heading and permalink for Definition 73">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-5"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">74</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-5-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> indicate <span class="process-math">\(A\text{,}\)</span> with <span class="process-math">\(\Pr(A) = p\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-5-1-1-4">
|
||||
\begin{gather*}
|
||||
E(X) = 0 \cdot \underbrace{\Pr(X = 0)}_{1 - p} + 1 \cdot \underbrace{\Pr(X = 1)}_{p} = 0(1-p) + 1p = p.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">That is: "E(indicator random variable) = Pr(event that it indicates)".</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 74"><a href="#notes-01-29-3-5" title="Copy heading and permalink for Example 74" aria-label="Copy heading and permalink for Example 74">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">75</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-01-29-3-6-1-1">Suppose we flip a fair coin 100 times. Let <span class="process-math">\(H_1, H_2, \dotsc, H_{100}\)</span> indicate heads on each flip. Then <span class="process-math">\(\Pr(H_i = 1) = \frac{1}{2}\text{,}\)</span> so <span class="process-math">\(E(H_i) = \frac{1}{2}\)</span> for every <span class="process-math">\(i\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="notes-01-29-3-6-1-2">
|
||||
<div class="para">What if <span class="process-math">\(X\)</span> 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:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-6-1-2-2">
|
||||
\begin{gather*}
|
||||
\Pr(X = 1) = \left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) = \frac{1}{16} = \E(X).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-6-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 75"><a href="#notes-01-29-3-6" title="Copy heading and permalink for Example 75" aria-label="Copy heading and permalink for Example 75">🔗</a></div></article><article class="definition definition-like" id="def-continuous-EV"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">76</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="def-continuous-EV-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X \in [a, b]\)</span> be a continuous random variable with pdf <span class="process-math">\(f(x)\text{.}\)</span> The <dfn class="terminology">expected value</dfn> of <span class="process-math">\(X\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="def-continuous-EV-1-1-5">
|
||||
\begin{gather*}
|
||||
\E(X) = \int_a^b x f(x)\ dx.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-continuous-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 76"><a href="#def-continuous-EV" title="Copy heading and permalink for Definition 76" aria-label="Copy heading and permalink for Definition 76">🔗</a></div></article><div class="para" id="notes-01-29-3-8">It’s worth putting this side-by-side with <a href="notes-01-29.html#def-discrete-EV" class="xref" data-knowl="./knowl/xref/def-discrete-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 73">Definition 73</a> 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.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="notes-01-29-3-9"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">77</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-9-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X \in [0, 1]\)</span> with pdf <span class="process-math">\(f(x) = \frac{3}{2}\sqrt{x}\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-9-1-1-3">
|
||||
\begin{align*}
|
||||
\E(X) \amp = \int_0^1 x f(x)\ dx \\
|
||||
\amp = \int_0^1 x \frac{3}{2} x^{1/2}\ dx \\
|
||||
\amp = \int_0^1 \frac{3}{2} x^{3/2}\ dx \\
|
||||
\amp = \frac{3}{2} \frac{2x^{5/2}}{5}\bigg|_0^1 \\
|
||||
\amp = \frac{3}{5} \cdot 1^{5/2} - \frac{3}{5} \cdot 0^{5/2} \\
|
||||
\amp = \frac{3}{5}.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">What would happen if you forgot the <span class="process-math">\(x\text{?}\)</span> Then the calculation would become:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-9-1-1-5">
|
||||
\begin{align*}
|
||||
\int_0^1 f(x)\ dx = 1 = \text{total probability} \amp
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">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!</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 77"><a href="#notes-01-29-3-9" title="Copy heading and permalink for Example 77" aria-label="Copy heading and permalink for Example 77">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-10"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">78</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-01-29-3-10-1-1">Find <span class="process-math">\(\E(X^2)\)</span> given a distribution for <span class="process-math">\(X\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-01-29-3-10-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">79<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(X\)</span><div class="autopermalink" data-description="Table 79: Distribution for X"><a href="#notes-01-29-3-10-1-2" title="Copy heading and permalink for Table 79: Distribution for X" aria-label="Copy heading and permalink for Table 79: Distribution for X">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">-1</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.4</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.3</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para" id="notes-01-29-3-10-1-3">We can start by writing a distribution table for <span class="process-math">\(X^2\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-10-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-01-29-3-10-1-4"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">80<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(X^2\)</span><div class="autopermalink" data-description="Table 80: Distribution for X^2"><a href="#notes-01-29-3-10-1-4" title="Copy heading and permalink for Table 80: Distribution for X^2" aria-label="Copy heading and permalink for Table 80: Distribution for X^2">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X^2 = x)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.5</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.2</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">9</td>
|
||||
<td class="l m b0 r0 l0 t0 lines">0.3</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="notes-01-29-3-10-1-5">
|
||||
<div class="para">Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-10-1-5-1">
|
||||
\begin{gather*}
|
||||
\E(X^2) 1(0.5) + 4(0.2) + 9(0.3) = 4.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-10-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 78"><a href="#notes-01-29-3-10" title="Copy heading and permalink for Example 78" aria-label="Copy heading and permalink for Example 78">🔗</a></div></article><article class="theorem theorem-like" id="notes-01-29-3-11"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">81</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-11-1-1">
|
||||
<div class="para">If <span class="process-math">\(X\)</span> takes on values <span class="process-math">\(x_1, \dotsc, x_n\)</span> with probabilities <span class="process-math">\(p_1, \dotsc, p_n\text{,}\)</span> and <span class="process-math">\(h\)</span> is any function, then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-11-1-1-5">
|
||||
\begin{gather*}
|
||||
\E(h(X)) = \sum_{i=1}^n h(x_i)p_i.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-11-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 81"><a href="#notes-01-29-3-11" title="Copy heading and permalink for Theorem 81" aria-label="Copy heading and permalink for Theorem 81">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-12"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">82</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-12-1-1">
|
||||
<div class="para">Revisiting the previous example, the theorem says we don’t need to first create the distribution table for <span class="process-math">\(X^2\text{.}\)</span> We can use the distribution table for <span class="process-math">\(X\text{,}\)</span> and just apply the square to each value:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-12-1-1-3">
|
||||
\begin{gather*}
|
||||
\E(X^2) = (-1)^2(0.4) + (1)^2(0.1) + (2)^2(0.2) + (3)^2(0.3) = 4.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 82"><a href="#notes-01-29-3-12" title="Copy heading and permalink for Example 82" aria-label="Copy heading and permalink for Example 82">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-13"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">83</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-13-1-1">
|
||||
<div class="para">The situation with continuous random variables is similar. Let <span class="process-math">\(X\in [0, 1]\)</span> with pdf <span class="process-math">\(f(x) = \frac{3}{2}\sqrt{x}\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-13-1-1-3">
|
||||
\begin{align*}
|
||||
\E(X^2) \amp = \int_0^1 x^2 f(x)\ dx \\
|
||||
\amp = \int_0^1 x^2 \frac{3}{2} x^{1/2}\ dx \\
|
||||
\amp = \int_0^1 \frac{3}{2} x^{5/2}\ dx \\
|
||||
\amp = \frac{3}{2}\cdot \frac{2x^{7/2}}{7}\bigg|_0^1 \\
|
||||
\amp = \frac{3}{7}\cdot 1^{7/2} - 0\cdot 0^{7/2} \\
|
||||
\amp = \frac{3}{7}.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Notice, in particular, that <span class="process-math">\(\E(X^2) \neq \E(X).\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 83"><a href="#notes-01-29-3-13" title="Copy heading and permalink for Example 83" aria-label="Copy heading and permalink for Example 83">🔗</a></div></article><article class="theorem theorem-like" id="thm-linearity"><h4 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">84</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="thm-linearity-1-1">
|
||||
<div class="para">If <span class="process-math">\(X, Y\)</span> are random variables with finite expected value and <span class="process-math">\(k \in \R\text{,}\)</span> then:</div>
|
||||
<div class="displaymath process-math" id="thm-linearity-1-1-3">
|
||||
\begin{align*}
|
||||
\E(X+Y) \amp = E(X) + E(Y) \\
|
||||
\E(kX) \amp = kE(X)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-linearity-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 84"><a href="#thm-linearity" title="Copy heading and permalink for Theorem 84" aria-label="Copy heading and permalink for Theorem 84">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-15"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">85</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-15-1-1">
|
||||
<div class="para">Let <span class="process-math">\(N\)</span> be the number of heads in <span class="process-math">\(n\)</span> coin flips with bias <span class="process-math">\(p\text{.}\)</span> Then <span class="process-math">\(N\sim \Bin(n, p)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-15-1-1-5">
|
||||
\begin{gather*}
|
||||
\E)N) = \sum_{i=0}^n i\cdot b(i; n, p) = \sum_{i = 0}^n i \cdot {n \choose i} p^i (1-p)^{n - i}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Gross.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-15-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="notes-01-29-3-15-1-2">
|
||||
<div class="para">Instead of calculating this directly, define <span class="process-math">\(H_1, H_2, \dotsc, H_n\)</span> to indicate heads on each flip. Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-15-1-2-2">
|
||||
\begin{align*}
|
||||
N \amp = H_1 + H_2 + \dotsb + H_n \\
|
||||
\E(N) \amp = \E(H_1 + H_2 + \dotsb + H_n) \\
|
||||
\E(N) \amp = \E(H_1) + \E(H_2) + \dotsb + \E(H_n) \\
|
||||
\amp = p + p + \dotsb + p \\
|
||||
\amp = np.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-15-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 85"><a href="#notes-01-29-3-15" title="Copy heading and permalink for Example 85" aria-label="Copy heading and permalink for Example 85">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-16"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">86</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-16-1-1">
|
||||
<div class="para">Flip a fair coin 100 times. Then, the expected number of heads is:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-16-1-1-1">
|
||||
\begin{gather*}
|
||||
\underbrace{(100)}_{n}\underbrace{(0.5)}_{p} = 50.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-16-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 86"><a href="#notes-01-29-3-16" title="Copy heading and permalink for Example 86" aria-label="Copy heading and permalink for Example 86">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-17"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">87</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-01-29-3-17-1-1">
|
||||
<div class="para">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 <span class="process-math">\(R_1, R_2, \dotsc, R_{97}\)</span> each indicating a run of 4 heads starting at the specified flip. Then <span class="process-math">\(\E(R_i) = \frac{1}{16}\)</span> for each <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(T\)</span> be the number of runs of 4 heads. Then:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-17-1-1-5">
|
||||
\begin{align*}
|
||||
\E(T) \amp = \E(R_1 + \dotsb + R_{97}) \\
|
||||
\amp = \E(R_1) + \dotsb + (R_{97}) \\
|
||||
\amp = \frac{1}{16} + \dotsb + \frac{1}{16} \\
|
||||
\amp = \frac{97}{16} \approx 6.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-17-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 87"><a href="#notes-01-29-3-17" title="Copy heading and permalink for Example 87" aria-label="Copy heading and permalink for Example 87">🔗</a></div></article><article class="example example-like" id="notes-01-29-3-18"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">88</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-01-29-3-18-1-1">Consider a geometric distribution <span class="process-math">\(T \sim \Geom(p)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-18-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-01-29-3-18-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">89<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Geometric Distribution<div class="autopermalink" data-description="Table 89: Geometric Distribution"><a href="#notes-01-29-3-18-1-2" title="Copy heading and permalink for Table 89: Geometric Distribution" aria-label="Copy heading and permalink for Table 89: Geometric Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
|
||||
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(T = k)\)</span></td>
|
||||
<td class="l m b1 r0 l0 t0 lines">with <span class="process-math">\(p = \frac{1}{2}\)</span>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">1</td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{2}\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">2</td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\((1-p)p\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{4}\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\((1-p)^2p\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{8}\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines">4</td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\((1-p)^3p\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{16}\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para" id="notes-01-29-3-18-1-3">Then <span class="process-math">\(\E(T) = (1)\left(\frac{1}{2}\right) + (2)\left(\frac{1}{4}\right) + (3)\left(\frac{1}{8}\right) + \dotsb\)</span> is an infinite summation.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-18-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="notes-01-29-3-18-1-4">
|
||||
<div class="para">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, <span class="process-math">\(T = 1\text{.}\)</span> In the second case, what is the average value of <span class="process-math">\(T\text{?}\)</span> Starting at flip 2, it will take on average <span class="process-math">\(\E(T)\)</span> flips to see heads. Since we already flipped the coin once, the total number of flips will be <span class="process-math">\(1 + \E(T)\text{.}\)</span> So, we can write:</div>
|
||||
<div class="displaymath process-math" id="notes-01-29-3-18-1-4-5">
|
||||
\begin{align*}
|
||||
\E(T) \amp = 1 \left(\frac{1}{2}\right) + (1 + \E(T)) \left(1 - \frac{1}{2}\right) \\
|
||||
\E(T) \amp = \frac{1}{2} + \frac{1}{2} + \frac{\E(T)}{2} \\
|
||||
\E(T) - \frac{\E(T)}{2} \amp = 1 \\
|
||||
\frac{\E(T)}{2} \amp = 1 \\
|
||||
\E(T) \amp = 2.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-29-3-18-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 88"><a href="#notes-01-29-3-18" title="Copy heading and permalink for Example 88" aria-label="Copy heading and permalink for Example 88">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Expected Value"><a href="#notes-01-29-3" title="Copy heading and permalink for Subsection: Expected Value" aria-label="Copy heading and permalink for Subsection: Expected Value">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Jan 29"><a href="#notes-01-29" title="Copy heading and permalink for Section: Thursday, Jan 29" aria-label="Copy heading and permalink for Section: Thursday, Jan 29">🔗</a></div></section></div>
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<a class="pretext-link" href="https://pretextbook.org" title="PreTeXt"><div class="logo"><svg xmlns="http://www.w3.org/2000/svg" height="100%" viewBox="338 3000 8772 6866" role="img"><title>PreTeXt logo</title><g style="stroke-width:.025in; stroke:currentColor; fill:none"><polyline points="472,3590 472,9732 " style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round; "></polyline><path style="stroke-width:126;stroke-linecap:butt;" d="M 4724,9448 A 4660 4660 0 0 1 8598 9259"></path><path style="stroke-width:174;stroke-linecap:butt;" d="M 4488,9685 A 4228 4228 0 0 0 472 9732"></path><path style="stroke-width:126;stroke-linecap:butt;" d="M 4724,3590 A 4241 4241 0 0 1 8598 3496"></path><path style="stroke-width:126;stroke-linecap:round;" d="M 850,3496 A 4241 4241 0 0 1 4724 3590"></path><path style="stroke-width:126;stroke-linecap:round;" d="M 850,9259 A 4507 4507 0 0 1 4724 9448"></path><polyline points="5385,4299 4062,8125" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="8598,3496 8598,9259" style="stroke-width:126; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="850,3496 850,9259" style="stroke-width:126; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="4960,9685 4488,9685" style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="3070,4582 1889,6141 3070,7700" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="6418,4582 7600,6141 6418,7700" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="8976,3590 8976,9732" style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round;"></polyline><path style="stroke-width:174;stroke-linecap:butt;" d="M 4960,9685 A 4228 4228 0 0 1 8976 9732"></path></g></svg></div></a><a class="runestone-link" href="https://runestone.academy" title="Runestone Academy"><img class="logo" src="https://runestone.academy/runestone/static/images/RAIcon_cropped.png" alt="Runstone Academy logo"></a><a class="mathjax-link" href="https://www.mathjax.org" title="MathJax"><img class="logo" src="https://www.mathjax.org/badge/badge-square-2.png" alt="MathJax logo"></a>
|
||||
</div>
|
||||
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|
||||
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|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
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|
||||
</ul>
|
||||
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|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
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|
||||
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|
||||
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|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
|
||||
+6
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@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-01-27.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
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<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-01-29.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
|
||||
\newcommand{\Z}{\mathbb Z}
|
||||
\newcommand{\Q}{\mathbb Q}
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
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|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
@@ -227,7 +231,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</h2>
|
||||
<nav class="summary-links"><ul><li><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></li></ul></nav><div class="autopermalink" data-description="Chapter: Quizzes"><a href="#quizzes" title="Copy heading and permalink for Chapter: Quizzes" aria-label="Copy heading and permalink for Chapter: Quizzes">🔗</a></div></section></div>
|
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<div id="ptx-content-footer" class="ptx-content-footer">
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<a class="previous-button button" href="notes-01-27.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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<a class="previous-button button" href="notes-01-29.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
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</div></main>
|
||||
</div>
|
||||
<div id="ptx-page-footer" class="ptx-page-footer">
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -288,7 +292,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">70</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
</h3>
|
||||
<div class="para logical" id="thm-FTC-2-1">
|
||||
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
|
||||
@@ -299,7 +303,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 70: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" data-description="Theorem 90: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 90: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 90: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -273,7 +277,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">70</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
|
||||
</h3>
|
||||
<div class="para logical" id="thm-FTC-2-1">
|
||||
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
|
||||
@@ -284,7 +288,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 70: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" data-description="Theorem 90: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 90: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 90: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
@@ -211,6 +211,10 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
Reference in New Issue
Block a user