The Method of Cumulative Distribution Function (Approach 1)

By definition, the cdf of Y is

F(Y) = P(Y≤y) - P(g(X) ≤y)

=∑ f _x ( x ), in the discrete case                                                       (6.1)

g(x) ≤y

= ∫fx ( x ) dx in the continuous case

g(x) ≤y

where fx ( x ) is the joint pmf or pdf of random vector X as the case may be. Thus distribution of Y can be obtained from that of X by using relation (6.1).

We consider some examples to illustrate the approach.

Example 1:

Find the cdf of Y = Z² where Z ^-N (0,l)

Solution:

F(y)= P(Y ≤ y) = p(z²≤y) = P(-√y≤ √z≤√y)

= Φ(√y)- Φ(-√y) = Φ(√y) - (1 - Φ√y)

Hence

F(Y) = [2 Φ(√y)-l]

where Φ ( z ) is the cdf of Z.