(4) (a) Suppose that X and Y are independent continuous random variables, with densities fx and fy respectively. Show that the cumulative distribution function of X + Y is given by •t P(X + Y ≤t) = LL - = [[ fx (x) fy (y = x) dx dy = [*[* ƒx(x − y) fy (y) dy dx. - Use the above equation to show that the density fx+y of X + Y is fx+y (t) = [ fx(x)fy(t = x) dx - = - = fx (t − y) fy (y) dy. (b) (c) 88 -∞ Suppose that X and Y are independent uniform (0, 2) random variables. Use the result in part (a), or otherwise (such as techniques in Section 5.1), compute the density of X + Y. Find an example of jointly continuous random variables U and V such that the marginal den- sities of U and V are both uniform(0, 1) distribution, but U + V must have a different density as in part (b).