Normal Distribution

$\begin{figure}\begin{center}\BoxedEPSF{NormalDistribution.epsf scaled 650}\end{center}\end{figure}$

Another name for a Gaussian Distribution. Given a normal distribution in a Variate with Mean $\mu$ and Variance $\sigma^2$ ,

$\begin{displaymath} P(x)\,dx={1\over \sigma\sqrt{2\pi}} e^{-(x-\mu)^2/2\sigma^2}\,dx, \end{displaymath}$

the so-called ``Standard Normal Distribution'' is given by taking $\mu=0$ and $\sigma^2=1$ . An arbitrary normal distribution can be converted to a Standard Normal Distribution by changing variables to $z\equiv (x-\mu)/\sigma$ , so $dz=dx/\sigma$ , yielding

$\begin{displaymath} P(x)\,dx = {1\over\sqrt{2\pi}} e^{-z^2/2}\,dz. \end{displaymath}$

The Fisher-Behrens Problem is the determination of a test for the equality of Means for two normal distributions with different Variances.