Question

6. (a) For normal distribution with mean zero and variance 2

σ show that:

2

(| |) σ

π

E x =

 (

Answer 1

Given,

Mean $\mu =\ 0\\$

Variance $= {\sigma}^2$

To prove: $\sigma=\sqrt{E(X^2)}$

We know that

$Var(X)= E(X^2)-E(X)^2$

$Var(X)=E(X^2)-\mu^2$

But $\mu = 0$

So, $Var(X)= E(X^2)\\\Rightarrow \sigma^2 = E(X^2)$

Hence, $\sigma = \sqrt{E(X^2)}$