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+\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} -- 2.47.1