45 lines
3.6 KiB
HTML
Executable File
45 lines
3.6 KiB
HTML
Executable File
<!DOCTYPE html>
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<html lang="en-US" dir="ltr">
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<meta name="robots" content="noindex, nofollow">
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</head>
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<body class="ignore-math">
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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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</body>
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</html>
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