A Random Variable g( X):

Given a r.v. X and a function g( x), the expression

defines a new r.v. Y. With y a given number, we denote D _y the subset of R _X (range of X) such that g( x) ? y. Then

where ( X ? D _y) is the event consisting of all outcomes ? such that the point X( ?) ? D _y. Hence,

If X is a continuous r.v. with pdf f _X( x), then

Determination of f _Y( y) from f _X( x):

Let X be a continuous r.v. with pdf f _X( x). If the transformation y = g( x) is one-to-one and has the inverse transformation

then the pdf of Y is given by (Prob. 6.30)

Note that if g( x) is a continuous monotonic increasing or decreasing function, then the transformation y = g( x) is one-to-one. If the transformation y = g( x) is not one-to-one, f _Y( y) is obtained as follows:

Denoting the real roots of y = g( x) by x _k, that is

then

where g ?( x) is the derivative of g( x).

B One Function of Two Random...