andy/Probability-and-Statistics-Notes
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Latest build deployed.

Browse Source
This commit is contained in:
andy
2026-08-01 15:00:14 +00:00
parent e43fb0d0ff
commit 331f133dfe
1278 changed files with 19041 additions and 18362 deletions
Download Patch File Download Diff File Expand all files Collapse all files
knowl/xref/def-binomial-distribution.html
Executable → Regular
+5 -4
View File
@@ -18,11 +18,12 @@
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<article class="definition definition-like"><h4 class="heading">
<article class="definition definition-like"><h2 class="heading">
<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>
</h4>
<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>
</h2>
<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>
</div>
<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>
<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>
</article><span class="incontext"><a class="internal" href="sec-Discrete-RVs.html#def-binomial-distribution">In Context</a></span>
</body>
</html>