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a\operatorname{E}(X)+b \\ + \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\ + \end{align*} + + \subsection*{Linear combination of two random variables} + + \begin{align*} + \textbf{Mean:} && \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\ + \textbf{Variance:} && \operatorname{Var}(aX+bY) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \tag{if \(X\) and \(Y\) are independent}\\ + \end{align*} + + \section{Sample mean} + + \[ \overline{x} = \dfrac{\Sigma x}{n} \] + + where \(n\) is the size of the sample (number of sample points) + + \subsubsection*{\colorbox{cas}{On CAS:}} + + \begin{enumerate} + \item Spreadsheet + \item In cell A1: \verb;mean(randNorm(sd, mean, sample size)); + \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range + \item Input range as A1:An where \(n\) is the number of samples + \item Graph \(\rightarrow\) Histogram + \end{enumerate} + + \subsubsection*{Sample size of \(n\)} + + \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \] + + Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) + + + +\end{document} -- 2.47.1