Question

Let Z be a standard unit normal random variable. Let W=Z^2 What is the probability density function of W? Find E[W] and Var(W)

Answer 1

Here we have given that Z is standard Normal variable.

That is Z ~

Let W =

We know the property that Square of standard normal distribution follows Chi-square distribution with

degrees of freedom = 1

Definition : The chi-squared distribution  with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.

Here we have given only one standard normal variable Z so, W= has Chi- square distribution with 1 degress of freedom.

We know that the property of Chi-square (   ) distribution that , If Y has chi-square distribution with k degrees of freedom then it's mean = k and variance = 2*k

Here for variable W = has chi-square distribution with degrees of freedom = 1 that is k = 1

So E(W) = 1 and Variance V(W) = 2

Probability density function of W is,

, where w>0