You have two lightbulbs for a particular lamp. Let X
the lifetime of the first bulb and Y
the lifetime of the second bulb (both in 1000s of hours). Suppose that X and Y are independent and that each has an exponential distribution with parameter 
1.

a. What is the joint pdf of X and Y?

b. What is the probability that each bulb lasts at most 1000 hours (i.e., X  1 and Y  1)?

c. What is the probability that the total lifetime of the two bulbs is at most 2? [Hint: Draw a picture of the region A
{(x, y): x  0, y  0, x y  2} before integrating.]

d. What is the probability that the total lifetime is between 1 and 2?