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Let Z~N(0;1) and Y=Z^2. Find the cumulative distribution function, probability density function, and moment generating function...

Let Z~N(0;1) and Y=Z^2. Find the cumulative distribution function, probability density function, and moment generating function of Y.

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Answer #1

Given .

Let .

The cumulative distribution function of Y is given by

Take . when z=0, u=0 and when

.

Hence the cumulative distribution function of Y is

since

Now the probability density function of Y is obtained by differentiating cumultive distribution function and is given by

which is the Chi square density with 1 degree of freedom.

The moment generating function is given by

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