\begin{align*}
\Pr(Y \le y) &= \Pr(aX+b \le y) \\
&= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
- &= \int^{\dfrac{y-b}{a}}_{-\infty} f(x) \> dx
+ &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
\end{align*}
\begin{align*}
where \(n\) is the size of the sample (number of sample points) and \(x\) is the value of a sample point
- \subsubsection*{\colorbox{cas}{On CAS:}}
+ \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
\begin{enumerate}
\item Spreadsheet
\item Input range as A1:An where \(n\) is the number of samples
\item Graph \(\rightarrow\) Histogram
\end{enumerate}
+ \end{tcolorbox}
\subsubsection*{Sample size of \(n\)}
Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
- \colorbox{cas}{On CAS:} Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left \\
- To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+ For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
+
+ \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
+
+ \begin{itemize}
+ \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
+ \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+ \end{itemize}
+ \end{tcolorbox}
\section{Normal distributions}
\[ Z = \frac{X - \mu}{\sigma} \]
- Normal distributions must have are (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
+ Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
\pgfmathdeclarefunction{gauss}{2}{%
\pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
-\begin{tikzpicture}
+{\begin{center} \begin{tikzpicture}
+ \pgfplotsset{set layers}
\begin{axis}[every axis plot post/.append style={
- mark=none,domain=-3:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
- axis x line*=bottom, % no box around the plot, only x and y axis
- axis y line*=left, % the * suppresses the arrow tips
+ mark=none,domain=-3:3,samples=50,smooth},
+ axis x line=bottom,
+ axis y line=left,
enlargelimits=upper,
- ytick={0.5},
+ x=\textwidth/10,
+ ytick={0.55},
yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)},
xtick={-2,-1,0,1,2},
- xticklabels={\(\mu-2\sigma\), \(\mu-\sigma\), \(\mu\), \(\mu+\sigma\), \(\mu+2\sigma\)},
+ x tick label style = {font=\footnotesize},
+ xticklabels={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
xlabel={\(x\)},
every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+ every axis y label/.style={at={(axis description cs:-0.02,0.2)}, anchor=south west, rotate=90},
ylabel={\(\Pr(X=x)\)}]
\addplot {gauss(0,0.75)};
\end{axis}
-\end{tikzpicture}
+\begin{axis}[every axis plot post/.append style={
+ mark=none,domain=-3:3,samples=50,smooth},
+ axis x line=bottom,
+ enlargelimits=upper,
+ x=\textwidth/10,
+ xtick={-2,-1,0,1,2},
+ axis x line shift=30pt,
+ hide y axis,
+ x tick label style = {font=\footnotesize},
+ xlabel={\(Z\)},
+ every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
+ \addplot {gauss(0,0.75)};
+\end{axis}
+\end{tikzpicture}\end{center}}
\section{Central limit theorem}