Let X and Y be two continuous random variables with probabil

Let X and Y be two continuous random variables with probability density functions fX (x) and fY (y).

(a) Determine an expression for the density of their difference, fXY , if X and Y are independent.

(b) Use your expression to compute the density for X Y for X Unif[0, 1] and Y Unif[0,1].

(c) What is the density of the sum X+Y for XUnif [0,1] and Y Unif [0,1]?

(d) How are the answers to parts (b) and (c) similar? What assumptions on X and Y make this so? Prove your observation is true by manipulating your expression in part (a).

Solution