1. Suppose that X and Y are independent binomial random variables with parameters (n, p) and (m, p). Argue probabilistically (no computations necessary) that X + Y is binomial with parameters (n + m, p).

2. Let X1, X2, X3, and X4 be independent continuous random variables with a com- mon distribution function F and let p = P{X1 <>X2 > X3 X4}

(a) Argue that the value of p is the same for all continuous distribution functions F.

(b) Find p by integrating the joint density function over the appropriate region.

(c) Find p by using the fact that all 4! possible orderings of X1, ... , X4 are equally likely.