From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 5 Aug 2019 01:41:14 +0000 (+1000)
Subject: [methods] Bernoulli distributions
X-Git-Tag: yr12~74
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/b643c6f2fb3884cbceacffc11198fc673e7eaf63

[methods] Bernoulli distributions
---

diff --git a/methods/statistics.pdf b/methods/statistics.pdf
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--- a/methods/statistics.tex
+++ b/methods/statistics.tex
@@ -7,6 +7,10 @@
 \usepackage{fancyhdr}
 \usepackage{pgfplots}
 \usepackage{tabularx}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{xcolor} % used only to show the phantomed stuff
+\definecolor{cas}{HTML}{e6f0fe}
 
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Unit 3 Methods Statistics}
@@ -116,4 +120,34 @@
     6&1&&6&&15&&20&&15&&6&&1
   \end{tabular}
 
+  \colorbox{cas}{On CAS:} (soft keys) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr);
+
+  \section{Binomial distributions}
+
+  (aka Bernoulli distributions)
+
+  \begin{align*}
+    \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\
+    &= {n \choose x} p^x q^{n-x}
+  \end{align*}
+
+  \begin{enumerate}
+    \item Two possible outcomes: \textbf{success} or \textbf{failure}
+    \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
+    \item Finite number \(n\) of independent trials
+  \end{enumerate}
+
+  If these conditions are met, then it is a Binomial Random Variable. This variable is said to have a \textit{binomial probability distribution}.
+
+  \begin{itemize}
+    \item \(n\) is the number of trials
+    \item There are two possible outcomes: \(S\) or \(F\)
+    \item \(\Pr(\text{success}) = p\)
+    \item \(\Pr(\text{failure}) = 1-p = q\)
+    \item Shorthand notation: \(X \sim \operatorname{Bi}(n,p)\)
+  \end{itemize}
+
+  \colorbox{cas}{On CAS:} Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf; \\
+  Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success)
+
 \end{document}