Question

Suppose that a random variable ( X ) has probability density function [ f_{X}(x)=frac{1}{sqrt{2 pi} sigma} e^{-frac{(x-mu)^{2}}{2 sigma^{2}}}, ext { where } mu, sigma ext { are constants and } sigma>0 ] Define ( Y=a X+b ) where ( a, b ) are constants and ( a>0 ). (a) Write down the distribution function of ( Y, F(y)=mathbb{P}(Y leq y) ), as an integration of ( f(x) ). (b) Apply the Fundamental Theorem of Calculus and find the probability density function ( f(y) ) of ( Y ). What is the expected value and standard deviation of ( Y ) ? (c) Find constants ( c ) and ( d ) such that the random variable ( Z=c X+d ) has expected value 0 and standard deviation 1. (In this case, ( Z ) is called to have the standard normal distribution.)

          Suppose that a random variable ( X ) has probability density function
[
f_{X}(x)=frac{1}{sqrt{2 pi} sigma} e^{-frac{(x-mu)^{2}}{2 sigma^{2}}}, 	ext { where } mu, sigma 	ext { are constants and } sigma>0
]
Define ( Y=a X+b ) where ( a, b ) are constants and ( a>0 ).
(a) Write down the distribution function of ( Y, F(y)=mathbb{P}(Y leq y) ), as an integration of ( f(x) ).
(b) Apply the Fundamental Theorem of Calculus and find the probability density function ( f(y) ) of ( Y ). What is the expected value and standard deviation of ( Y ) ?
(c) Find constants ( c ) and ( d ) such that the random variable ( Z=c X+d ) has expected value 0 and standard deviation 1. (In this case, ( Z ) is called to have the standard normal distribution.)

$Suppose that a random variable ( X ) has probability density function [ fX(x)=frac1sqrt2 pi sigma e^-frac(x-mu)^22 sigma^2, ext where mu, sigma ext are constants and sigma>0 ] Define ( Y=a X+b ) where ( a, b ) are constants and ( a>0 ). (a) Write down the distribution function of ( Y, F(y)=mathbbP(Y leq y) ), as an integration of ( f(x) ). (b) Apply the Fundamental Theorem of Calculus and find the probability density function ( f(y) ) of ( Y ). What is the expected value and standard deviation of ( Y ) ? (c) Find constants ( c ) and ( d ) such that the random variable ( Z=c X+d ) has expected value 0 and standard deviation 1. (In this case, ( Z ) is called to have the standard normal distribution.)$

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