Find the distribution of W = X2 when

(a) X is N(0, 4),

(b) X is N(0, σ2).

Answer :

Step 1 of 3 :

A random variable X is normally distributed with mean and standard deviation .

The probability density function is

f(x) =

The claim is to find the distribution of w = when.

X ~ N(0, 4)X ~ N(0, )In general, ~ with one degrees of freedom.

Where, v =

The probability density function of v is

P( V v) = dy , 0 < v <

= dy

Step 2 of 3 :

Here, X follows N(0,4)w = , v = it follows with one degrees of freedom.

Let w =

= 4v

Then, P(W w) = P(4v w)

= P(v )

= dy - (1)

Let, z = 4y

= dy

Substitute in equation 1

=

=

Which follows gamma distribution with = ½ and = 8

Therefore,

P(W w) = be the distribution function of w = when

X ~ N(0, 4).