1. (Distribution of one function of two continuous random variables.) Let X and Y be continuous...

1. (Distribution of one function of two continuous random variables.) Let X and Y be continuous random variables with joint density the density for X + Y, is given by

Follow the procedure given. to obtain the joint density for (U, V). Integrate the joint density to obtain the marginal density for U.

2. Let X and Y be independent standard normal random variables. Let U = X + Y. Use Exercise 48 to prove that U follows a normal distribution with mean 0 and variance 2

Exercise 48

(Distribution of one function of two continuous random variables.) Let X and Y be continuous random variables with joint density the density for X + Y, is given by

Follow the procedure given. to obtain the joint density for (U, V). Integrate the joint density to obtain the marginal density for U.