Let X and Y be independent Exponential random variables withcommon mean 1.

Their joint pdf is f(x,y) = exp (-x-y) for x > 0 and y > 0, f(x, y ) = 0 otherwise. (See "Independence" on page 349)

Let U = min(X, Y) and V = max (X, Y).

The joint pdf of U and V is f(u, v) = 2 exp (-u-v) for 0 < u< v < infinity, f(u, v ) = 0 otherwise. WORDS: f(u, v ) istwice f(x, y) above the diagonal in the first quadrant, otherwisef(u, v ) is zero.

(a). Use the "Marginals" formula on page 349 to get the marginalpdf f(u) of U from joint pdf f(u, v) HINT: You should know theanswer before you plug into the formula.

(b) Use the "Marginals" formula on page 349 to get the marginalpdf f(v) of V from joint pdf f(u, v) HINT: You found f(v) in aprevious HW by finding the CDF of V. You can also figure out theanswer by thinking about two independent light bulbs and adding theprobabilities of the two ways that V can fall into a tiny intervaldv.

(c) Find the conditional pdf of V, given that U = 2. (See page411). HINT: You can figure out what the answer has to be bythinking about two independent light bulbs and remembering thememoryless property.

(d) Find P( V > 3 | U= 2 ). (See bottom of page 411. Do theappropriate integral, but you should know what the answer willbe.)

(e) Find the conditional pdf of U, given that V = 1. (See page411).

(f) Find P ( U < 0.5 | V = 1).

HINT: You should know ahead of time whether the answer is >or < or = 1/2.