Please take the time to post the question rather than attaching it as a word document.

The question is (more or less):

X is distributed as a Poisson random variable with the usual pmf. Find the pmf of Y when:

(a) Y = 4X

(b) Y = X^2.

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In each case the following theorem can be used:

Suppose that X is a discrete random variable with pmf f(x). Let Y = u(X) define aone-to-onetransformation between the value of X and the value of Y so that the equation y = u(u) can be uniquely solved for x in terms of y, x = v(y), say. Then the pmf of Y is g(y) = f(w(y)).

Note that in (b) the necessary condition is satisfied because . Also, is OK (why?).