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Show histogram with Histogram key in top left \\ + To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable + + \section{Normal distributions} + + mean = mode = median + + \[ Z = \frac{X - \mu}{\sigma} \] + + Normal distributions must have are (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{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 + enlargelimits=upper, + ytick={0.5}, + yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)}, + xtick={-2,-1,0,1,2}, + xticklabels={\(\mu-2\sigma\), \(\mu-\sigma\), \(\mu\), \(\mu+\sigma\), \(\mu+2\sigma\)}, + xlabel={\(x\)}, + every axis x label/.style={at={(current axis.right of origin)},anchor=north west}, + ylabel={\(\Pr(X=x)\)}] + \addplot {gauss(0,0.75)}; +\end{axis} +\end{tikzpicture} + + \section{Central limit theorem} + + If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\). \end{document} -- 2.47.1