4-67. Consider the problem of obtaining the probability distribution of a random variable Y from information about the probability distribution of a random variable X, where y = g(x) is a functional relationship between the values X and Y. To perform the indicated change of variable or transformation we shall utilize the following THEOREM. Let the probability density function of the random variableXbe given by f (x) and let the function y = g(x) define a one-to-one transformation betweenXand Y. (A one-to-one transformation implies that g is either an increasing or decreasing function for all admissible x.)

In addition, let the unique inverse transformation of g be denoted as x = w(y) and let dx/dy = w(y) be continuous and not vanish for all admissible y’s. Then the probability density function of Y is given by h(y) =

dw(y)/dy

f w(y) , dw(y)/dy
= 0.

Here 

dw(y)/dy

denotes the absolute value of dw(y)/dy. If f (x) = e−x, x ≥ 0, and y = g(x) = 3x, find h(y).