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Functions of a Random Variable

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Adding Constant to Random Variables

~X is a random variable and _ ~Z = ~X + ~c , _ ~c &in. &reals. .

F_~Z ( ~z ) _ = _ P ( ~X + ~c < ~z ) _ = _ F_~X ( ~z - ~c )

f_~Z ( ~z ) _ = _ F#~'_~Z ( ~z ) _ = _ F#~'_~X ( ~z - ~c ) _ = _ f_~X ( ~z - ~c )

So the density and distribution functions are simply translated ~c units to the right (~c > 0) or left (~c < 0). (See diagram right.)

Scalar Product of Random Variables

~X is a random variable and _ ~Z = &alpha. ~X , _ &alpha. &in. &reals. .

F_~Z ( ~z ) _ = _ P ( &alpha.~X < ~z )

Case 1) _ &alpha. > 0

F_~Z ( ~z ) _ = _ P ( ~X < ~z ./ &alpha. ) _ = _ F_~X ( ~z ./ &alpha. )

f_~Z ( ~z ) _ = _ fract{d F_~X ( ~z ./ &alpha. ), d~z} _ = _ fract{1, &alpha.} F#'_~X ( ~z ./ &alpha. ) _ = _ fract{1, &alpha.} f_~X ( ~z ./ &alpha. )

Case 2) _ &alpha. < 0

F_~Z ( ~z ) _ = _ P ( ~X > ~z ./ &alpha. ) _ = _ 1 - F_~X ( ~z ./ &alpha. )

f_~Z ( ~z ) _ = _ - fract{d F_~X ( ~z ./ &alpha. ), d~z} _ = _ - fract{1, &alpha.} F#'_~X ( ~z ./ &alpha. ) _ = _ - fract{1, &alpha.} f_~X ( ~z ./ &alpha. )

Expressing this generally we write: _ _ f_~Z ( ~z ) _ = _ fract{1, |&alpha.|} f_~X ( ~z ./ &alpha. )

Square of Random Variables

~X is a random variable and _ ~Z = ~X ^2. _ If ~z < 0, then

F_~Z ( ~z ) _ = _ P ( ~X ^2 < ~z ) _ = _ 0

If ~z > 0 then

F_~Z ( ~z ) _ = _ P ( ~X ^2 < ~z ) _ = _ P ( | ~X | < &sqrt.$~z ) _ = _ P ( - &sqrt.$~z < ~X < &sqrt.$~z )

_ _ _ _ _ = _ F_~X ( &sqrt.$~z ) _ - _ F_~X ( - &sqrt.$~z )

f_~Z (~z) _ = _ fract{1,2&sqrt.$~z} F#'_~X ( &sqrt.$~z ) _ - _ fract{- 1,2&sqrt.$~z}F#'_~X ( - &sqrt.$~z ) _ = _ fract{1,2&sqrt.$~z} rndb{f_~X ( &sqrt.$~z ) _ + _ f_~X ( - &sqrt.$~z )}

~{Example}

For the the density function is symmetric about zero, i.e.

f_~X ( - ~x) _ = _ f_~X ( ~x ) _ = _ fract{1,&sqrt.${2&pi.}} exp rndb{fract{- ~x^2,2 }}

so if _ ~X ~ N( 0 , 1 ) _ and _ ~Z = ~X ^2

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

This is the density function for a distribution with one degree of freedom.

Variance of Standard Normal Distribution

We defined the of a continuous random variable, and saw that:

var( ~X ) _ = _ E( ~X ^2 ) - E( ~X )^2

For the we saw that E ( ~X ) = 0 , so _ _ var( ~X ) _ = _ E( ~X ^2 )

But, as shown above _ ~X ^2 ~ &chi. ( 1 ) , _ and the random variable with one degree of freedom is 1, i.e.

~X ~ N ( 0 , 1 ) , _ then _ _ var ( ~X ) = 1.