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Executable → Regular
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para">Suppose an event occurs with probability <span class="process-math">\(p\text{.}\)</span> If we perform <span class="process-math">\(n\)</span> independent trials, let <span class="process-math">\(S\)</span> be the random variable which counts the number of trials in which the event occurred. Then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> We’ll write <span class="process-math">\(S \sim \Bin(n, p)\)</span> to denote this. For each value <span class="process-math">\(0 \leq k \leq n\text{,}\)</span> we’ll write <span class="process-math">\(b(k)\)</span> for <span class="process-math">\(\Pr(S = k)\text{.}\)</span> (If we want to keep track of the parameter values, we may write <span class="process-math">\(b(k; n, p)\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#def-binomial-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</h2>
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<div class="para">Suppose an event occurs with probability <span class="process-math">\(p\text{.}\)</span> If we perform <span class="process-math">\(n\)</span> independent trials, let <span class="process-math">\(S\)</span> be the random variable which counts the number of trials in which the event occurred. Then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n\)</span> and <span class="process-math">\(p\text{.}\)</span> We’ll write <span class="process-math">\(S \sim \Bin(n, p)\)</span> to denote this. For each value <span class="process-math">\(0 \leq k \leq n\text{,}\)</span> we’ll write <span class="process-math">\(b(k)\)</span> for <span class="process-math">\(\Pr(S = k)\text{.}\)</span> (If we want to keep track of the parameter values, we may write <span class="process-math">\(b(k; n, p)\text{.}\)</span>)<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-binomial-distribution-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 2.1.6"><a href="#def-binomial-distribution" title="Copy heading and permalink for Definition 2.1.6" aria-label="Copy heading and permalink for Definition 2.1.6">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#def-binomial-distribution">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Definition 2.1.6"><a tabindex="-1" href="#def-binomial-distribution" title="Copy heading and permalink for Definition 2.1.6" aria-label="Copy heading and permalink for Definition 2.1.6">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#def-binomial-distribution">In Context</a></span>
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Executable → Regular
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para logical">
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<div class="para">Let <span class="process-math">\(X\)</span> take values in the interval <span class="process-math">\([a, b]\text{.}\)</span> We say that <span class="process-math">\(X\)</span> has the <dfn class="terminology">uniform distribution</dfn> if, for any subinterval <span class="process-math">\([c, d] \subset [a, b]\text{,}\)</span> the probability that <span class="process-math">\(X\)</span> lies within <span class="process-math">\([c, d]\)</span> is equal to the proportion of the total sample space taken by <span class="process-math">\([c, d]\text{.}\)</span> That is:</div>
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<div class="displaymath process-math">
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<div class="para">In this case, <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span>
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#def-continuous-uniform-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-continuous-uniform-distribution-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 2.2.2"><a href="#def-continuous-uniform-distribution" title="Copy heading and permalink for Definition 2.2.2" aria-label="Copy heading and permalink for Definition 2.2.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-continuous-uniform-distribution">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Definition 2.2.2"><a tabindex="-1" href="#def-continuous-uniform-distribution" title="Copy heading and permalink for Definition 2.2.2" aria-label="Copy heading and permalink for Definition 2.2.2">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-continuous-uniform-distribution">In Context</a></span>
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Executable → Regular
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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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 the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\text{.}\)</span> The <dfn class="terminology">expected value</dfn> (also called <dfn class="terminology">mean</dfn>, or <dfn class="terminology">expectation</dfn>) of <span class="process-math">\(X\)</span> is the weighted average of the values of <span class="process-math">\(X\text{,}\)</span> where the weights are the probabilities of <span class="process-math">\(X\)</span> taking each value:</div>
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<div class="displaymath process-math">
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\E(X) = \sum_{i=1}^n x_i \Pr(X = x_i).
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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 class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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 3.1.1"><a href="#def-discrete-EV" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#def-discrete-EV">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Definition 3.1.1"><a tabindex="-1" href="#def-discrete-EV" title="Copy heading and permalink for Definition 3.1.1" aria-label="Copy heading and permalink for Definition 3.1.1">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#def-discrete-EV">In Context</a></span>
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Executable → Regular
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.8</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para logical">
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<div class="para">Consider the parametrized family of functions:</div>
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<div class="displaymath process-math">
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@@ -30,9 +30,11 @@ f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigm
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</div>
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<div class="para">The function <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is called a <dfn class="terminology">normal density function</dfn> with parameters <span class="process-math">\(\mu\)</span> and <span class="process-math">\(\sigma^2\text{.}\)</span> A continuous random variable <span class="process-math">\(X\)</span> with pdf <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is said to have a <dfn class="terminology">normal distribution</dfn>. We’ll write <span class="process-math">\(X \sim \Norm(\mu, \sigma^2)\text{.}\)</span>
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<div class="autopermalink" data-description="Paragraph"><a href="#def-normal-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <div class="para">In the specific case that <span class="process-math">\(\mu = 0\)</span> and <span class="process-math">\(\sigma^2 = 1\text{,}\)</span> we call the resulting distribution the <dfn class="terminology">standard normal distribution</dfn>. We’ll use the notation <span class="process-math">\(\phi(x)\)</span> for the standard normal density function, and <span class="process-math">\(\Phi(x)\)</span> for the standard normal cumulative distribution function.<div class="autopermalink" data-description="Paragraph"><a href="#def-normal-distribution-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-normal-distribution-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="para">In the specific case that <span class="process-math">\(\mu = 0\)</span> and <span class="process-math">\(\sigma^2 = 1\text{,}\)</span> we call the resulting distribution the <dfn class="terminology">standard normal distribution</dfn>. We’ll use the notation <span class="process-math">\(\phi(x)\)</span> for the standard normal density function, and <span class="process-math">\(\Phi(x)\)</span> for the standard normal cumulative distribution function.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-normal-distribution-1-2" 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 2.2.8"><a href="#def-normal-distribution" title="Copy heading and permalink for Definition 2.2.8" aria-label="Copy heading and permalink for Definition 2.2.8">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-normal-distribution">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Definition 2.2.8"><a tabindex="-1" href="#def-normal-distribution" title="Copy heading and permalink for Definition 2.2.8" aria-label="Copy heading and permalink for Definition 2.2.8">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-normal-distribution">In Context</a></span>
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Executable → Regular
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para logical">
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<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. The <dfn class="terminology">variance</dfn> of <span class="process-math">\(X\)</span> is:</div>
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<div class="displaymath process-math">
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\Var(X) = \E\left[ \left(X - \E(X)\right)^2 \right].
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\end{gather*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#def-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-variance-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 3.2.1"><a href="#def-variance" title="Copy heading and permalink for Definition 3.2.1" aria-label="Copy heading and permalink for Definition 3.2.1">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#def-variance">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Definition 3.2.1"><a tabindex="-1" href="#def-variance" title="Copy heading and permalink for Definition 3.2.1" aria-label="Copy heading and permalink for Definition 3.2.1">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Variance.html#def-variance">In Context</a></span>
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Executable → Regular
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para logical">
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<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
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\amp = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
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\end{align*}
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<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-EV-die-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="Example 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-die">in-context</a></span>
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<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.2"><a tabindex="-1" href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-die">In Context</a></span>
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Executable → Regular
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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</h2>
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<div class="para">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-EV-sum-2-dice-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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<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.1.4: "><a tabindex="-1" href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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<tr class="header-horizontal">
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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
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</tr>
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</table></div></figure> <div class="para">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</table></div>
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</figure>
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<div class="para">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
||||
<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-sum-2-dice">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.3"><a tabindex="-1" href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-sum-2-dice">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h4 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Suppose <span class="process-math">\(S \sim \Bin(n, p)\text{.}\)</span> <a href="sec-Discrete-RVs.html#def-binomial-distribution" class="xref" data-knowl="./knowl/xref/def-binomial-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.1.6">Definition 2.1.6</a> describes a binomially distributed random variable as counting the number of occurrences of some event, which either happens or not in each of <span class="process-math">\(n\)</span> independent trials. This leads us to a very natural idea: identify each individual possible occurrence of the event, and assign it an indicator random variable. In this case, let <span class="process-math">\(H_i\)</span> indicate that flip <span class="process-math">\(i\)</span> comes up heads. Then</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/def-binomial-distribution.html ./knowl/xref/example-indicator-EV.html">
|
||||
@@ -38,8 +38,9 @@ S = H_1 + H_2 + \dotsb + H_n.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Remember this formula! We’ll make frequent use of it.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-binomial-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-binomial-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="Example 3.1.11"><a href="#example-binomial-EV" title="Copy heading and permalink for Example 3.1.11" aria-label="Copy heading and permalink for Example 3.1.11">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-binomial-EV">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.11"><a tabindex="-1" href="#example-binomial-EV" title="Copy heading and permalink for Example 3.1.11" aria-label="Copy heading and permalink for Example 3.1.11">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-binomial-EV">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h3 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Recall the setup from <a href="sec-Expected-Value.html#example-binomial-EV" class="xref" data-knowl="./knowl/xref/example-binomial-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.11">Example 3.1.11</a>: a binomial random variable <span class="process-math">\(S\)</span> is the sum <span class="process-math">\(H_1 + H_2 + \dotsb + H_n\)</span> of <span class="process-math">\(n\)</span> indicator random variables, each with parameter <span class="process-math">\(p\text{.}\)</span> The results of different coin flips are independent from each other, so the random variables <span class="process-math">\(H_1, \dotsc, H_n\)</span> are independent from each other. From <a href="sec-Variance.html#example-indicator-variance" class="xref" data-knowl="./knowl/xref/example-indicator-variance.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.2.6">Example 3.2.6</a>, we know that <span class="process-math">\(\Var(H_i) = p(1-p)\)</span> for each <span class="process-math">\(i\text{.}\)</span> Therefore:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-EV.html ./knowl/xref/example-indicator-variance.html">
|
||||
@@ -32,8 +32,9 @@
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Remember this formula! We’ll make frequent use of it.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-binomial-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-binomial-variance-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 3.2.7"><a href="#example-binomial-variance" title="Copy heading and permalink for Example 3.2.7" aria-label="Copy heading and permalink for Example 3.2.7">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#example-binomial-variance">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.2.7"><a tabindex="-1" href="#example-binomial-variance" title="Copy heading and permalink for Example 3.2.7" aria-label="Copy heading and permalink for Example 3.2.7">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Variance.html#example-binomial-variance">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+17
-10
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h3 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Suppose we observe a cell and measure the time <span class="process-math">\(T\)</span> until a toxin molecule leaves the cell. Then <span class="process-math">\(T \sim \Exp(\lambda)\)</span> for some unknown rate parameter <span class="process-math">\(\lambda\text{:}\)</span>
|
||||
</div>
|
||||
@@ -38,9 +38,11 @@ L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3t} (-0.3) \\
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">The critical point is <span class="process-math">\(\lambda = \frac{1}{0.3} \approx 3.33\text{.}\)</span> Since <span class="process-math">\(\L' \gt 0\)</span> on <span class="process-math">\((0, 3.33)\)</span> and <span class="process-math">\(\L' \lt 0\)</span> on <span class="process-math">\((3.33, \infty)\text{,}\)</span> there is a global maximum at <span class="process-math">\(\lambda \approx 3.33\text{,}\)</span> which is therefore the MLE.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para">The specific time <span class="process-math">\(0.3\)</span> minutes doesn’t particularly matter in this calculation. Whatever the time <span class="process-math">\(t\text{,}\)</span> essentially the same calculation will result in a MLE of <span class="process-math">\(\lambda = 1/t\text{.}\)</span> But what if we collect multiple pieces of data?<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Time Data<div class="autopermalink" data-description="Table 4.1.12: Time Data"><a href="#example-exponential-MLE-1-3" title="Copy heading and permalink for Table 4.1.12: Time Data" aria-label="Copy heading and permalink for Table 4.1.12: Time Data">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-exponential-MLE-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para">The specific time <span class="process-math">\(0.3\)</span> minutes doesn’t particularly matter in this calculation. Whatever the time <span class="process-math">\(t\text{,}\)</span> essentially the same calculation will result in a MLE of <span class="process-math">\(\lambda = 1/t\text{.}\)</span> But what if we collect multiple pieces of data?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-exponential-MLE-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Time Data<div class="autopermalink" aria-hidden="true" data-description="Table 4.1.12: Time Data"><a tabindex="-1" href="#example-exponential-MLE-1-3" title="Copy heading and permalink for Table 4.1.12: Time Data" aria-label="Copy heading and permalink for Table 4.1.12: Time Data">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines">Molecule</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Waiting Time</td>
|
||||
@@ -71,8 +73,11 @@ L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3t} (-0.3) \\
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.9\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1.11\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para">How should take all of this data into account in our maximum likelihood estimation? We might consider taking the average of all of the separate rate estimations, which would give <span class="process-math">\(1.87\text{.}\)</span> Is this the most likely? We need some mathematical justification.<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical">
|
||||
</table></div>
|
||||
</figure>
|
||||
<div class="para">How should take all of this data into account in our maximum likelihood estimation? We might consider taking the average of all of the separate rate estimations, which would give <span class="process-math">\(1.87\text{.}\)</span> Is this the most likely? We need some mathematical justification.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-exponential-MLE-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical">
|
||||
<div class="para">To account for multiple, independent data points, we should multiply the probability densities for each in the creation of our likelihood function:</div>
|
||||
<div class="displaymath process-math">
|
||||
\begin{align*}
|
||||
@@ -90,9 +95,11 @@ L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3t} (-0.3) \\
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">The only critical point is <span class="process-math">\(5/3.1 \approx 1.61\text{.}\)</span> Since <span class="process-math">\(\L' \gt 0\)</span> on <span class="process-math">\((0, 1.61)\)</span> and <span class="process-math">\(\L' \lt 0\)</span> on <span class="process-math">\((1.61, \infty)\text{,}\)</span> there is a global maximum at <span class="process-math">\(\lambda = 1.61\text{,}\)</span> which is therefore the MLE.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para">It may seem less clear how to generalize this calculation for other tables of data. Observe that the value <span class="process-math">\(3.1\)</span> is the sum of the five times in the table, so <span class="process-math">\(3.1/5\)</span> is the average time. The MLE turned out to be the reciprocal of the average time (just as the MLE with only one data point was the reciprocal of that one time). Notice that this does <em class="emphasis">not</em> match the guess we made previously of averaging the individual rate estimations for each data point.<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-exponential-MLE-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para">It may seem less clear how to generalize this calculation for other tables of data. Observe that the value <span class="process-math">\(3.1\)</span> is the sum of the five times in the table, so <span class="process-math">\(3.1/5\)</span> is the average time. The MLE turned out to be the reciprocal of the average time (just as the MLE with only one data point was the reciprocal of that one time). Notice that this does <em class="emphasis">not</em> match the guess we made previously of averaging the individual rate estimations for each data point.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-exponential-MLE-1-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 4.1.11"><a href="#example-exponential-MLE" title="Copy heading and permalink for Example 4.1.11" aria-label="Copy heading and permalink for Example 4.1.11">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Likelihood.html#example-exponential-MLE">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 4.1.11"><a tabindex="-1" href="#example-exponential-MLE" title="Copy heading and permalink for Example 4.1.11" aria-label="Copy heading and permalink for Example 4.1.11">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Likelihood.html#example-exponential-MLE">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h4 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -29,8 +29,9 @@
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">That is, the expected value of an indicator random variable is the probability of the event that it indicates.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-indicator-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="Example 3.1.5"><a href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-indicator-EV">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.5"><a tabindex="-1" href="#example-indicator-EV" title="Copy heading and permalink for Example 3.1.5" aria-label="Copy heading and permalink for Example 3.1.5">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-indicator-EV">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h3 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> indicate an event <span class="process-math">\(A\)</span> which has probability <span class="process-math">\(p\text{.}\)</span> In <a href="sec-Expected-Value.html#example-indicator-EV" class="xref" data-knowl="./knowl/xref/example-indicator-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.5">Example 3.1.5</a>, we saw <span class="process-math">\(\E(X) = p\text{.}\)</span> Notice that <span class="process-math">\(X\)</span> takes the values 0 and 1 (with probabilities <span class="process-math">\(1 - p\)</span> and <span class="process-math">\(p\text{,}\)</span> respectively). But <span class="process-math">\(0^2 = 0\text{,}\)</span> and <span class="process-math">\(1^1 = 1\text{,}\)</span> so the expected value calculation for <span class="process-math">\(X^2\)</span> is precisely the same as for <span class="process-math">\(X\text{!}\)</span>
|
||||
</div>
|
||||
@@ -35,8 +35,9 @@
|
||||
\Var(X) \amp \E\left(X^2\right) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p).
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-indicator-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-indicator-variance-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 3.2.6"><a href="#example-indicator-variance" title="Copy heading and permalink for Example 3.2.6" aria-label="Copy heading and permalink for Example 3.2.6">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#example-indicator-variance">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.2.6"><a tabindex="-1" href="#example-indicator-variance" title="Copy heading and permalink for Example 3.2.6" aria-label="Copy heading and permalink for Example 3.2.6">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Variance.html#example-indicator-variance">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+11
-7
@@ -18,11 +18,12 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h3 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para">A poll asks two yes/no questions. Let <span class="process-math">\(X\)</span> indicate a yes on Question 1 and <span class="process-math">\(Y\)</span> indicate a yes on Question 2. When the data is collected, the following joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is created:<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-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"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution for Indicator Random Variables<div class="autopermalink" data-description="Table 2.3.3: Joint Distribution for Indicator Random Variables"><a href="#example-joint-indicators-1-2" title="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables" aria-label="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
</h2>
|
||||
<div class="para">A poll asks two yes/no questions. Let <span class="process-math">\(X\)</span> indicate a yes on Question 1 and <span class="process-math">\(Y\)</span> indicate a yes on Question 2. When the data is collected, the following joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is created:<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-joint-indicators-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"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution for Indicator Random Variables<div class="autopermalink" aria-hidden="true" data-description="Table 2.3.3: Joint Distribution for Indicator Random Variables"><a tabindex="-1" href="#example-joint-indicators-1-2" title="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables" aria-label="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||||
@@ -38,7 +39,9 @@
|
||||
<td class="c m b0 r1 l0 t0 lines">0.3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.4</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical">
|
||||
</table></div>
|
||||
</figure>
|
||||
<div class="para logical">
|
||||
<div class="para">Note that the sum of all values in the table is 1; this is a probability distribution, and total probability must be 1. This table doesn’t show the distributions for <span class="process-math">\(X\)</span> or <span class="process-math">\(Y\)</span> individually. The data for the two distributions is mixed together. We can, if we wish, take this information and determine separate distributions for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span> For the distribution for <span class="process-math">\(X\text{,}\)</span> we need to know <span class="process-math">\(\Pr(X = 0)\)</span> and <span class="process-math">\(\Pr(X = 1)\)</span> (with no reference to <span class="process-math">\(Y\)</span>). We can get these probabilities by summing along the columns of the table:</div>
|
||||
<div class="displaymath process-math">
|
||||
\begin{align*}
|
||||
@@ -53,8 +56,9 @@
|
||||
\Pr(Y = 1) \amp = 0.3 + 0.4 = 0.7
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-joint-indicators-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 2.3.2"><a href="#example-joint-indicators" title="Copy heading and permalink for Example 2.3.2" aria-label="Copy heading and permalink for Example 2.3.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Joint-Distributions.html#example-joint-indicators">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 2.3.2"><a tabindex="-1" href="#example-joint-indicators" title="Copy heading and permalink for Example 2.3.2" aria-label="Copy heading and permalink for Example 2.3.2">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Joint-Distributions.html#example-joint-indicators">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+8
-6
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h4 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Suppose we observe traffic along a particular stretch of highway which typically has 200 cars pass per hour. Let <span class="process-math">\(N\)</span> be the number of cars seen in a 2-hour observation period. Since the parameter <span class="process-math">\(\lambda = 200\)</span> cars per hour is given for a 1-hour period, we should scale it to find the rate for a 2-hour period: <span class="process-math">\(\lambda t = (200)(2) = 400\)</span> cars per 2-hours. Then, for example:</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -28,8 +28,9 @@
|
||||
\Pr(N = 375) = p(375; 200, 2) = \frac{400^{375}}{375!} e^{-400} \approx 0.01.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-poisson-rate-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical">
|
||||
<div class="para">Now consider the random variable <span class="process-math">\(M\)</span> counting the number of cars seen in a 30-minute observation period. We should not calculate <span class="process-math">\(\lambda t = (200)(30) = 6000\text{,}\)</span> since <span class="process-math">\(\lambda\)</span> measures time in hours and <span class="process-math">\(t\)</span> measures time in minutes. We should first convert <span class="process-math">\(t = 0.5\)</span> hours, then we can find our appropriately scaled rate information: <span class="process-math">\((200)(0.5) = 100\)</span> cars per half-hour. Then, for example:</div>
|
||||
<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
@@ -37,8 +38,9 @@
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Note that when we write the notation <span class="process-math">\(p(k; \lambda, t)\text{,}\)</span> we assume that <span class="process-math">\(\lambda\)</span> and <span class="process-math">\(t\)</span> are expressed with the same time units already.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-poisson-rate-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-poisson-rate-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 2.1.17"><a href="#example-poisson-rate" title="Copy heading and permalink for Example 2.1.17" aria-label="Copy heading and permalink for Example 2.1.17">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#example-poisson-rate">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 2.1.17"><a tabindex="-1" href="#example-poisson-rate" title="Copy heading and permalink for Example 2.1.17" aria-label="Copy heading and permalink for Example 2.1.17">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#example-poisson-rate">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+8
-6
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h4 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">An experiment consists of rolling a fair 6-sided die two times. Let <span class="process-math">\(A\)</span> be the event that the sum of the rolls is at least 10. To find <span class="process-math">\(\Pr(A)\text{,}\)</span> we note that <span class="process-math">\(A = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\}\text{,}\)</span> which has 6 elements. Since <span class="process-math">\(|\Omega| = 36\text{,}\)</span> we have</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -28,8 +28,9 @@
|
||||
\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{36} = \frac{1}{6}\text{.}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-rolls-conditional-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical">
|
||||
<div class="para">Let <span class="process-math">\(B\)</span> be the event that the first roll is 6. If we pause after the first die roll seeing the value of 6, we might be more inclined to expect a sum of at least 10. The evidence that we’ve already seen changes our understanding of the situation. Since <span class="process-math">\(|B| = 6\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Also, <span class="process-math">\(A\cap B = \{(6, 4), (6, 5), (6, 6)\}\text{,}\)</span> so <span class="process-math">\(\Pr(A\cap B) = \frac{3}{36} = \frac{1}{12}\text{.}\)</span> Finally:</div>
|
||||
<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
@@ -37,8 +38,9 @@
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Before the experiment, we would have said there was only a <span class="process-math">\(\frac{1}{6}\)</span> chance that the sum of the rolls is at least 10. However, with the additional knowledge of seeing the first roll of 6, we find the probability of a sum of at least 10 to be <span class="process-math">\(\frac{1}{2}\text{,}\)</span> substantially more likely than before.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-conditional-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-rolls-conditional-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 1.3.2"><a href="#example-rolls-conditional" title="Copy heading and permalink for Example 1.3.2" aria-label="Copy heading and permalink for Example 1.3.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Conditional-Probability.html#example-rolls-conditional">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 1.3.2"><a tabindex="-1" href="#example-rolls-conditional" title="Copy heading and permalink for Example 1.3.2" aria-label="Copy heading and permalink for Example 1.3.2">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Conditional-Probability.html#example-rolls-conditional">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+14
-9
@@ -18,13 +18,16 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h3 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para">What would the sample space look like if we roll the die two times and recorded the results?<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</h2>
|
||||
<div class="para">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-sample-space-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para">What would the sample space look like if we roll the die two times and recorded the results?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-sample-space-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="solutions">
|
||||
<details class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
|
||||
<div class="para logical">
|
||||
<div class="displaymath process-math">
|
||||
\begin{align*}
|
||||
@@ -38,10 +41,12 @@
|
||||
</div>
|
||||
<div class="para">Note that, for example, <span class="process-math">\((1, 2)\)</span> is a different outcome from <span class="process-math">\((2, 1)\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-sample-space-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="Answer 1.1.2.1"><a href="#example-sample-space-2" title="Copy heading and permalink for Answer 1.1.2.1" aria-label="Copy heading and permalink for Answer 1.1.2.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Example 1.1.2"><a href="#example-sample-space" title="Copy heading and permalink for Example 1.1.2" aria-label="Copy heading and permalink for Example 1.1.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Set-Theory.html#example-sample-space">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Answer 1.1.2.1"><a tabindex="-1" href="#example-sample-space-2" title="Copy heading and permalink for Answer 1.1.2.1" aria-label="Copy heading and permalink for Answer 1.1.2.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 1.1.2"><a tabindex="-1" href="#example-sample-space" title="Copy heading and permalink for Example 1.1.2" aria-label="Copy heading and permalink for Example 1.1.2">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Set-Theory.html#example-sample-space">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="example example-like"><h4 class="heading">
|
||||
<article class="example example-like"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.7</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be uniform on the interval <span class="process-math">\([a, b]\text{.}\)</span> So <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span> We might reasonably expect that the average value of <span class="process-math">\(X\)</span> would be the midpoint of the interval. Let’s check that:</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -33,8 +33,9 @@
|
||||
\amp = \frac{a + b}{2}.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#example-uniform-EV-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#example-uniform-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="Example 3.1.7"><a href="#example-uniform-EV" title="Copy heading and permalink for Example 3.1.7" aria-label="Copy heading and permalink for Example 3.1.7">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-uniform-EV">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 3.1.7"><a tabindex="-1" href="#example-uniform-EV" title="Copy heading and permalink for Example 3.1.7" aria-label="Copy heading and permalink for Example 3.1.7">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-uniform-EV">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,11 +18,12 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="fact theorem-like"><h3 class="heading">
|
||||
<article class="fact theorem-like"><h2 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.8</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">Binomial MLE.</span>
|
||||
</h3>
|
||||
<div class="para">If we see <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> coin flips, then the MLE of the bias <span class="process-math">\(p\)</span> is <span class="process-math">\(\frac{k}{n}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#fact-MLE-binomial-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</h2>
|
||||
<div class="para">If we see <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> coin flips, then the MLE of the bias <span class="process-math">\(p\)</span> is <span class="process-math">\(\frac{k}{n}\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#fact-MLE-binomial-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="Fact 4.1.8: Binomial MLE"><a href="#fact-MLE-binomial" title="Copy heading and permalink for Fact 4.1.8: Binomial MLE" aria-label="Copy heading and permalink for Fact 4.1.8: Binomial MLE">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Likelihood.html#fact-MLE-binomial">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Fact 4.1.8: Binomial MLE"><a tabindex="-1" href="#fact-MLE-binomial" title="Copy heading and permalink for Fact 4.1.8: Binomial MLE" aria-label="Copy heading and permalink for Fact 4.1.8: Binomial MLE">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Likelihood.html#fact-MLE-binomial">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
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@@ -18,17 +18,18 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="fact theorem-like"><h4 class="heading">
|
||||
<article class="fact theorem-like"><h2 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.12</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">Binomial Probability Formula.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
b(k; n, p) = {n \choose k} p^k (1-p)^{n-k}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-binomial-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#fact-binomial-probability-formula-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="Fact 2.1.12: Binomial Probability Formula"><a href="#fact-binomial-probability-formula" title="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#fact-binomial-probability-formula">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Fact 2.1.12: Binomial Probability Formula"><a tabindex="-1" href="#fact-binomial-probability-formula" title="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.12: Binomial Probability Formula">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#fact-binomial-probability-formula">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
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@@ -18,17 +18,18 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="fact theorem-like"><h4 class="heading">
|
||||
<article class="fact theorem-like"><h2 class="heading">
|
||||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.14</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">Geometric Probability Formula.</span>
|
||||
</h4>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="displaymath process-math">
|
||||
\begin{gather*}
|
||||
g(k; p) = (1-p)^{k-1} p
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#fact-geometric-probability-formula-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#fact-geometric-probability-formula-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="Fact 2.1.14: Geometric Probability Formula"><a href="#fact-geometric-probability-formula" title="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#fact-geometric-probability-formula">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Fact 2.1.14: Geometric Probability Formula"><a tabindex="-1" href="#fact-geometric-probability-formula" title="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula" aria-label="Copy heading and permalink for Fact 2.1.14: Geometric Probability Formula">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#fact-geometric-probability-formula">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+3
-2
@@ -18,7 +18,7 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">B.1.1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\chi^2\)</span> Critical Values<div class="autopermalink" data-description="Table B.1.1: \chi^2 Critical Values"><a href="#table-Chi-squared" title="Copy heading and permalink for Table B.1.1: \chi^2 Critical Values" aria-label="Copy heading and permalink for Table B.1.1: \chi^2 Critical Values">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">B.1.1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\chi^2\)</span> Critical Values<div class="autopermalink" aria-hidden="true" data-description="Table B.1.1: \chi^2 Critical Values"><a tabindex="-1" href="#table-Chi-squared" title="Copy heading and permalink for Table B.1.1: \chi^2 Critical Values" aria-label="Copy heading and permalink for Table B.1.1: \chi^2 Critical Values">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines">Degrees of Freedom</th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines">Critical value for <span class="process-math">\(p = 0.05\)</span>
|
||||
@@ -64,6 +64,7 @@
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(10\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(18.307\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure><span class="incontext"><a class="internal" href="app-Chi-squared-table.html#table-Chi-squared">in-context</a></span>
|
||||
</table></div>
|
||||
</figure><span class="incontext"><a class="internal" href="app-Chi-squared-table.html#table-Chi-squared">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+3
-2
@@ -18,7 +18,7 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Bin(3, 0.4)\)</span><div class="autopermalink" data-description="Table 2.1.8: \Bin(3, 0.4)"><a href="#table-binomial-example" title="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)" aria-label="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.1.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="process-math">\(\Bin(3, 0.4)\)</span><div class="autopermalink" aria-hidden="true" data-description="Table 2.1.8: \Bin(3, 0.4)"><a tabindex="-1" href="#table-binomial-example" title="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)" aria-label="Copy heading and permalink for Table 2.1.8: \Bin(3, 0.4)">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
|
||||
@@ -39,6 +39,7 @@
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.064</td>
|
||||
</tr>
|
||||
</table></div></figure><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#table-binomial-example">in-context</a></span>
|
||||
</table></div>
|
||||
</figure><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#table-binomial-example">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="theorem theorem-like"><h3 class="heading">
|
||||
<article class="theorem theorem-like"><h2 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Suppose <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">iid</dfn>) random variables with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Let:</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -37,8 +37,9 @@ S_n \approx \Norm(n \mu, n \sigma^2) \\
|
||||
A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-CLT-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#thm-CLT-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 4.2.1"><a href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-CLT.html#thm-CLT">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Theorem 4.2.1"><a tabindex="-1" href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-CLT.html#thm-CLT">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Executable → Regular
+5
-4
@@ -18,9 +18,9 @@
|
||||
<meta name="robots" content="noindex, nofollow">
|
||||
</head>
|
||||
<body class="ignore-math">
|
||||
<article class="theorem theorem-like"><h3 class="heading">
|
||||
<article class="theorem theorem-like"><h2 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable. Then:</div>
|
||||
<div class="displaymath process-math">
|
||||
@@ -28,8 +28,9 @@
|
||||
\Var(X) = \E\left(X^2\right) - \left(\E(X)\right)^2.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-variance-formula-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#thm-variance-formula-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 3.2.2"><a href="#thm-variance-formula" title="Copy heading and permalink for Theorem 3.2.2" aria-label="Copy heading and permalink for Theorem 3.2.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Variance.html#thm-variance-formula">in-context</a></span>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Theorem 3.2.2"><a tabindex="-1" href="#thm-variance-formula" title="Copy heading and permalink for Theorem 3.2.2" aria-label="Copy heading and permalink for Theorem 3.2.2">🔗</a></div>
|
||||
</article><span class="incontext"><a class="internal" href="sec-Variance.html#thm-variance-formula">In Context</a></span>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user