# Normal Distribution

Page Contents

## Distribution

~Z is said to have the #~{standard normal} or #~{standard Gaussian distribution} if

f( ~z ) _ = _ fract{1,&sqrt.\${2&pi.}} exp rndb{fract{- ~z^2,2 }}

We write _ ~Z ~ N(0,1).

~X is said to have the #~{normal} or #~{Gaussian distribution} with mean &mu. and variance &sigma.&powtwo. if

f( ~x ) _ = _ fract{1,&sqrt.\${2&pi.&sigma.&powtwo.}} exp rndb{fract{- ( ~x - &mu. )^2,2&sigma.&powtwo. }}

We write _ ~X ~ N( &mu. , &sigma.&powtwo. ). _ Note that _ ( ~X - &mu. ) ./ &sigma. ~ N(0,1).

There is no explicit formula for the distribution function of a normal distribution, this is obtained by integrating the density function.

## Density Function

The following result is useful in showing that the above formula is a proper density function:

int{,-&infty.,&infty.,}e^{-~y ^2 ./ 2} d~y _ = _ &sqrt.\${2&pi.}

#{Proof:}

int{,-&infty.,&infty.,}e^{-~y ^2 ./ 2} d~y _ # _ int{,-&infty.,&infty.,}e^{-~x ^2 ./ 2} d~x _ = _ int{,-&infty.,&infty.,}e^{-~x ^2 ./ 2} rndb{ int{,-&infty.,&infty.,}e^{-~y ^2 ./ 2} d~y } d~x

_ _ _ _ _ = _ int{,-&infty.,&infty.,}int{,-&infty.,&infty.,}e^{-( ~x ^2 + ~y ^2 ) ./ 2} d~x d~y

which in is

_ = _ int{,0,2&pi.,}int{,0,&infty.,}~r e^{-~r ^2 ./ 2} d~r d&theta. _ = _ int{,0,2&pi.,}script{sqrb{-e^{-~r ^2 ./ 2}},,,&infty.,0} d&theta. _ = _ int{,0,2&pi.,}1 d&theta. _ = _ 2&pi.

Polar Coordinates

~r ^2 _ = _ ~x ^2 + ~y ^2 , _ _ tan &theta. _ = _ ~y ./ ~x

&theta. _ = _ tan^{-1} ( ~y ./ ~x ) _ _ => _ _ d&theta. ./ d~y _ = _ 1 ./ ( ~x ^2 + ~y ^2 ) _ = _ 1 ./ ~r ^2 _ _ => _ _ d~y _ = _ ~r ^2 d&theta.

2~r ( d~r /d~x ) _ = _ 2~x _ _ => _ _ d~x _ = ~r d~r ./ ~x

## Standard Normal Distribution

We have shown that _ fract{1,&sqrt.\${2&pi.}} exp rndb{fract{- ~z^2,2 }} _ is a valid distribution function, which we call the standard normal distribution.

If _ ~Z ~ N ( 0 , 1 ) _ then _

E ~Z _ = _ fract{1,&sqrt.\${2&pi.}} int{,{-&infty.},{&infty.},} ~z exp rndb{fract{- ~z^2,2 }} _ d~z _ = _ 0

Since the integrand is anti-symmetric, i.e. _ ~{&integ.}__{-&infty.}^^0 _ = _ - ~{&integ.}__0^^{&infty.} _ _ so _ _ ~{&integ.}__{-&infty.}^^{&infty.} _ = _ ~{&integ.}__{-&infty.}^^0 _ + _ ~{&integ.}__0^^{&infty.} _ = _ 0