Question

Let X and Y be independent Exponential random variables with common 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 ) is twice f(x, y) above the diagonal in the first quadrant, otherwise f(u, v ) is zero.

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

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

(c) Find the conditional pdf of V, given that U = 2. (See page 411). HINT: You can figure out what the answer has to be by thinking about two independent light bulbs and remembering the memoryless property.

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

(e) Find the conditional pdf of U, given that V = 1. (See page 411).

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

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

Answer 1