Distribution of a Function of a random Variable- Part 1

Transformations of Variables of the Discrete Type
An alternative method of finding the distribution of a function of one or more random variables is called the change of variable technique..
There are some delicate questions (with particular reference to random
variables of the continuous type) involved in this technique, and these
make it desirable for us first to consider special cases. Let Ibea random variable of the discrete type, having
p.d.f. f(x). Let s/ denote the set of discrete points, at each of which
f(x) > 0, and let y = u(x) define a one-to-one transformation that maps
A onto B. If we solve y = u(x) for x in terms of y, say, x = w(y), then
for each y ? B, we have x = w(y) ) ? A. Consider the random variable
Y = u(X). If y ? B, then x = w(y) ? A, and the events Y = y [or u{X) = y] and X = w{y) are equivalent. Accordingly, the p.d.f. of Y is
g(y) = Pr (Y = y) = Pr [X = u/(y)] = /[w(y)], y ? B,

= 0 elsewhere.
Transformations of Variables of the Continuous Type
In the preceding section we introduced the notion of a one-to-one transformation and the mapping of a set A onto a set B under that transformation. Those ideas were sufficient to enable us to find the distribution of a function of several random variables o f the discrete
type. In this section we shall examine the same problem when the random variables are of the continuous type. It is again helpful to begin with a special problem.