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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-hw1-review" class="internal"><span class="title">HW 1 Q5</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
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<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
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<ul class="structural toc-item-list active"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-19.html" class="internal"><span class="title">Thursday, Feb 19</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-19.html#notes-02-19-3" class="internal"><span class="title">Central Limit Theorem</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-24.html" class="internal"><span class="title">Tuesday, Feb 24</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-3" class="internal"><span class="title">More Central Limit Theorem</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-4" class="internal"><span class="title">Confidence Intervals</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-10"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Tuesday, Feb 10</span>
|
||
</h2>
|
||
<section class="introduction" id="notes-02-10-2"><div class="para" id="notes-02-10-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div></section><section class="subsection" id="notes-02-10-3"><h3 class="heading hide-type">
|
||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Likelihood</span>
|
||
</h3>
|
||
<div class="para" id="notes-02-10-3-2">In probability theory, we start with some probability model and its parameter values, and we try to find the probabilities of seeing certain types of data. In statistics, we start with the collected data, and we try to find the most likely values of parameters for some underlying probability model.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="notes-02-10-3-3"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">106</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="notes-02-10-3-3-1-1">An <dfn class="terminology">estimator</dfn> is a way of estimating a parameter value based on data collected.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 106"><a href="#notes-02-10-3-3" title="Copy heading and permalink for Definition 106" aria-label="Copy heading and permalink for Definition 106">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-4"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">107</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="notes-02-10-3-4-1-1">
|
||
<div class="para">Suppose we flip a coin 100 times and see 52 heads. Let <span class="process-math">\(p\)</span> be the bias of the coin. Then we might estimate:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-4-1-1-2">
|
||
\begin{gather*}
|
||
\widehat{p} = \frac{52}{100}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">(The notation <span class="process-math">\(\widehat{p}\)</span> is sometimes used to indicate an estimator for <span class="process-math">\(p\text{.}\)</span>)</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 107"><a href="#notes-02-10-3-4" title="Copy heading and permalink for Example 107" aria-label="Copy heading and permalink for Example 107">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-5"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">108</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="notes-02-10-3-5-1-1">If we see <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> flips, then the estimator <span class="process-math">\(\widehat{p} = \frac{k}{n}\)</span> is called the <dfn class="terminology">common sense estimator</dfn> for the binomial distribution parameter <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 108"><a href="#notes-02-10-3-5" title="Copy heading and permalink for Definition 108" aria-label="Copy heading and permalink for Definition 108">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-6"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">109</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="notes-02-10-3-6-1-1">An estimator <span class="process-math">\(\widehat{p}\)</span> is <dfn class="terminology">unbiased</dfn> if <span class="process-math">\(\E(\widehat{p}) = p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 109"><a href="#notes-02-10-3-6" title="Copy heading and permalink for Definition 109" aria-label="Copy heading and permalink for Definition 109">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-7"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">110</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="notes-02-10-3-7-1-1">
|
||
<div class="para">Flip a coin 100 times, and let <span class="process-math">\(S\)</span> be the number of heads. Then <span class="process-math">\(S \sim \Bin(100, p)\text{.}\)</span> Let <span class="process-math">\(\widehat{p} = \frac{S}{100} = \frac{S}{n}\text{.}\)</span> Then:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-7-1-1-4">
|
||
\begin{align*}
|
||
\E(\widehat{p}) \amp = \E\left(\frac{S}{n}\right) \\
|
||
\amp = \frac{1}{n} \E\left(S\right) \\
|
||
\amp = \frac{1}{n} (np) \\
|
||
\amp = p.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">So <span class="process-math">\(\E(\widehat{p} = p\text{,}\)</span> i.e., the common sense estimator is unbiased.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 110"><a href="#notes-02-10-3-7" title="Copy heading and permalink for Example 110" aria-label="Copy heading and permalink for Example 110">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-8"><h4 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">111</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="notes-02-10-3-8-1-1">
|
||
<div class="para">We perform an experiment and collect data. Let <span class="process-math">\(p\)</span> be an unknown parameter value. The <dfn class="terminology">likelihood function</dfn> is:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-8-1-1-3">
|
||
\begin{gather*}
|
||
\L(p) = \Pr(\text{data} \mid \text{paramater value is } p).
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 111"><a href="#notes-02-10-3-8" title="Copy heading and permalink for Definition 111" aria-label="Copy heading and permalink for Definition 111">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-9"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">112</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="notes-02-10-3-9-1-1">
|
||
<div class="para">Let <span class="process-math">\(S \sim \Bin(100, p)\text{.}\)</span> Suppose we see 52 heads. Then:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-9-1-1-2">
|
||
\begin{align*}
|
||
\Pr(S = k) \amp = b(k; 100, p) = {100 \choose k}p^k (1 - p)^{100 - k} \\
|
||
\L(p) \amp = {100 \choose 52}p^{52} (1 - p)^{48}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">In the first line, the variable <span class="process-math">\(k\)</span> represents the data. In the second line, the variable <span class="process-math">\(p\)</span> represents the parameter value. Our goal, given the collected data, is to find the <dfn class="terminology">maximum likelihood estimation</dfn> (MLE) for the parameter value.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para logical" id="notes-02-10-3-9-1-2">
|
||
<div class="para">
|
||
<span class="process-math">\(\L(p)\)</span> is a continuous function over a closed interval <span class="process-math">\(p \in [0, 1]\text{,}\)</span> so we use the Closed Interval Method.</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-9-1-2-3">
|
||
\begin{align*}
|
||
\L'(p) \amp = {100 \choose 52}\left[ 52 p^{51} (1 - p)^{48} + p^{52}48 (1-p)^{47}(-1)\right] \\
|
||
\amp = {100 \choose 52} p^{51} (1-p)^{47}\left[ 52 (1 - p) - 48p \right] \\
|
||
\amp = {100 \choose 52} p^{51} (1-p)^{47}\left[ 52 - 100p \right]
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Now, we look for critical numbers in the interior of the interval:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-9-1-2-4">
|
||
\begin{gather*}
|
||
\L'(p) = 0 \text{ when } p = \frac{52}{100}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Finally, we test the critical numbers and the endpoints of the interval to find the max:</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="notes-02-10-3-9-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">113<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Check Candidate Locations for Max<div class="autopermalink" data-description="Table 113: Check Candidate Locations for Max"><a href="#notes-02-10-3-9-1-3" title="Copy heading and permalink for Table 113: Check Candidate Locations for Max" aria-label="Copy heading and permalink for Table 113: Check Candidate Locations for Max">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\L(p)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(52/100\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\gt 0\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
</tr>
|
||
</table></div></figure> <div class="para" id="notes-02-10-3-9-1-4">So <span class="process-math">\(\widehat{p} = \frac{52}{100}\)</span> is the MLE.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="notes-02-10-3-9-1-5">More generally, a similar calculation will show that, with <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> flips, the MLE will be <span class="process-math">\(\widehat{p} = \frac{k}{n}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 112"><a href="#notes-02-10-3-9" title="Copy heading and permalink for Example 112" aria-label="Copy heading and permalink for Example 112">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-10"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">114</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="notes-02-10-3-10-1-1">
|
||
<div class="para">Suppose we observe a cell, measuring 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 <span class="process-math">\(\lambda\text{,}\)</span> with pdf</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-4">
|
||
\begin{gather*}
|
||
f(t) =\lambda e^{-\lambda t}, \lambda \gt 0
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">If we see a toxin molecule leave at 0.3 min, what’s the MLE for <span class="process-math">\(\lambda\text{?}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-6">
|
||
\begin{gather*}
|
||
\L(\lambda) = \lambda e^{-0.3 \lambda} \quad \text{(density, not probability)}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">We want to maximize <span class="process-math">\(\L(\lambda)\)</span> over <span class="process-math">\(\lambda \in (0, \infty)\text{,}\)</span> an open interval. So we’ll use the "Open Interval Method".</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-9">
|
||
\begin{align*}
|
||
\L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3\lambda}(-0.3) \\
|
||
\amp = e^{-0.3\lambda}\left( 1 - 0.3 e^{-0.3\lambda}\right)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Then <span class="process-math">\(\L'(\lambda) = 0\)</span> when <span class="process-math">\(\lambda = \frac{1}{0.3}\approx 3.33\text{.}\)</span> Checking <span class="process-math">\(\lambda\)</span> values to the left and the right:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-13">
|
||
\begin{align*}
|
||
\L'(1) \amp = (+)(+) = (+) \\
|
||
\L'(10) \amp = (+)(-) = (-)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">So <span class="process-math">\(\L'(\lambda) \gt 0\)</span> (and therefore <span class="process-math">\(\L(\lambda)\)</span> is increasing) on <span class="process-math">\((0, 1/0.3)\text{,}\)</span> and <span class="process-math">\(\L'(\lambda) \lt 0\)</span> (and therefore <span class="process-math">\(\L(\lambda)\)</span> is decreasing) on <span class="process-math">\((1/0.3, \infty)\text{.}\)</span> Now we can conclude that <span class="process-math">\(\widehat{\lambda} = 1/0.3 \approx 3.33\)</span> is the location of a global (and not just local) maximum value.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="notes-02-10-3-10-1-2">More generally, if the observed time is <span class="process-math">\(t\text{,}\)</span> then the MLE will be <span class="process-math">\(\widehat{\lambda} = \frac{1}{t}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="notes-02-10-3-10-1-3">What if we had more data points? For example, suppose two toxin molecules leave the cell at <span class="process-math">\(t_1 = 0.3\)</span> min and <span class="process-math">\(t_2 = 0.5\)</span> min?<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="notes-02-10-3-10-1-4"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">115<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Waiting Times<div class="autopermalink" data-description="Table 115: Waiting Times"><a href="#notes-02-10-3-10-1-4" title="Copy heading and permalink for Table 115: Waiting Times" aria-label="Copy heading and permalink for Table 115: Waiting Times">🔗</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">Time</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Rate Estimate</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/0.3 \approx 3.33\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.5\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/0.5 = 2\)</span></td>
|
||
</tr>
|
||
</table></div></figure> <div class="para logical" id="notes-02-10-3-10-1-5">
|
||
<div class="para">How do we combine these data points? We could take the average of the rate estimates:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-1">
|
||
\begin{gather*}
|
||
\frac{3.33 + 2}{2} \approx 2.67
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Alternatively, we could average the times first, then create a new rate estimate from the average time:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-2">
|
||
\begin{align*}
|
||
\frac{0.3 + 0.5}{2} \amp = 0.4 \\
|
||
\frac{1}{0.4} \amp = 2.5
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Both of these make some sense, but let’s do a careful computation to be certain which way is correct (if either of them is!).</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-3">
|
||
\begin{align*}
|
||
\L(\lambda) \amp = \left( \lambda e^{-0.3 \lambda} \right)\left( \lambda e^{-0.5 \lambda} \right) \\
|
||
\amp = \lambda^2 e^{-0.3 \lambda - 0.5 \lambda} \\
|
||
\amp = \lambda^2 e^{- 0.8 \lambda}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Using the Open Interval Method:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-4">
|
||
\begin{align*}
|
||
\L'(\lambda) \amp = 2\lambda e^{-0.8 \lambda} + \lambda^2 e^{-0.8 \lambda} (-0.8) \\
|
||
\amp = \lambda e^{-0.8 \lambda} \left[ 2 - 0.8\lambda \right]
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">So <span class="process-math">\(\L'(\lambda) = 0\)</span> when <span class="process-math">\(\lambda = \frac{2}{0.8} = 2.5\text{.}\)</span> Testing points to the left and right:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-7">
|
||
\begin{align*}
|
||
\L'(1) \amp = (+)(+)(+) = (+) \\
|
||
\L'(5) \amp = (+)(+)(-) = (-)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Now <span class="process-math">\(\L'(\lambda) \gt 0\)</span> (<span class="process-math">\(\L(\lambda)\)</span> is increasing) on <span class="process-math">\((0, 2.5)\text{,}\)</span> and <span class="process-math">\(\L'(\lambda) \lt 0\)</span> (<span class="process-math">\(\L(\lambda)\)</span> is decreasing) on <span class="process-math">\((2.5, \infty)\text{.}\)</span> Therefore, the MLE is <span class="process-math">\(\widehat{\lambda} = \frac{2}{0.8} = 2.5\text{.}\)</span>
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para logical" id="notes-02-10-3-10-1-6">
|
||
<div class="para">Tracing the values <span class="process-math">\(2\)</span> and <span class="process-math">\(0.8\)</span> throughout the calculation, we can see that the value <span class="process-math">\(2\)</span> will generally match the number of waiting times collected, and the value <span class="process-math">\(0.8\)</span> will be the sum of the waiting times. So, generally, with collected waiting times of <span class="process-math">\(t_1, \dotsc, t_n\text{,}\)</span> the MLE will be:</div>
|
||
<div class="displaymath process-math" id="notes-02-10-3-10-1-6-6">
|
||
\begin{gather*}
|
||
\widehat{\lambda} = \frac{n}{t_1 + \dotsb + t_n} = \frac{1}{\text{avg time}}.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 114"><a href="#notes-02-10-3-10" title="Copy heading and permalink for Example 114" aria-label="Copy heading and permalink for Example 114">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Likelihood"><a href="#notes-02-10-3" title="Copy heading and permalink for Subsection: Likelihood" aria-label="Copy heading and permalink for Subsection: Likelihood">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Feb 10"><a href="#notes-02-10" title="Copy heading and permalink for Section: Tuesday, Feb 10" aria-label="Copy heading and permalink for Section: Tuesday, Feb 10">🔗</a></div></section></div>
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