\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}
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)
+
+ \subsection{Applications of binomial distributions}
+
+ \[ \Pr(X \ge a) = 1 - \Pr(X < a) \]
+
\end{document}