5-33. Let X,Y be independent random variables with probability mass functions f (X), g(Y), respectively (or probability density functions f (x), g(y), respectively). Let Z = X + Y with probability density function h(z).

Then:

(a) If X,Y are discrete, h(Z) = X f (X)g(Z − X)

(b) If X,Y are continuous, h(z) = 4

+∞

−∞ f (x)g(z − x) dx Suppose we have the following two independent probability density functions for the continuous random variables X and Y, respectively;

f (x) = -e−x, x > 0;

0 elsewhere;

g(y) = -e−y, y > 0;

0 elsewhere.

Let Z = X + Y. Use part

(b) to determine h(z).