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<article class="definition definition-like"><h4 class="heading">
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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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</h4>
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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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\begin{gather*}
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\Pr(c \leq X \leq d) = \frac{d - c}{b - a}.
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\end{gather*}
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</div>
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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 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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<article class="example example-like"><h4 class="heading">
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<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>
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</h4>
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<div class="para logical">
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<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>
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<div class="displaymath process-math">
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\begin{gather*}
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\Pr(N = 375) = p(375; 200, 2) = \frac{400^{375}}{375!} e^{-400} \approx 0.01.
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\end{gather*}
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</div>
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<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>
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</div> <div class="para logical">
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<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>
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<div class="displaymath process-math">
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\begin{gather*}
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\Pr(M = 110) = p(110; 200, 0.5) = \frac{100^{110}}{110!}e^{-100} \approx 0.02.
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\end{gather*}
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</div>
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<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>
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<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>
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</div>
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<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>
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<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>
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<div class="para logical">
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<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>
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<div class="displaymath process-math">
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\begin{gather*}
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\Pr(A) = \frac{|A|}{|\Omega|} = \frac{6}{36} = \frac{1}{6}\text{.}
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\end{gather*}
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</div>
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<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>
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</div> <div class="para logical">
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<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>
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<div class="displaymath process-math">
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\begin{gather*}
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\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{1/12}{1/6} = \frac{1}{2}
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\end{gather*}
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</div>
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<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>
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</div>
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<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>
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<article class="fact theorem-like"><h4 class="heading">
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<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>
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<div class="para logical">
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<div class="displaymath process-math">
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\begin{gather*}
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b(k; n, p) = {n \choose k} p^k (1-p)^{n-k}
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\end{gather*}
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</div>
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<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>
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</div>
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<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>
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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">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">
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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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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">0</td>
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<td class="c m b0 r0 l0 t0 lines">0.216</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">1</td>
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<td class="c m b0 r0 l0 t0 lines">0.432</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">2</td>
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<td class="c m b0 r0 l0 t0 lines">0.288</td>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">3</td>
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<td class="c m b0 r0 l0 t0 lines">0.064</td>
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</tr>
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</table></div></figure><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#table-binomial-example">in-context</a></span>
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