You have two lightbulbs for a particular lamp. Let X = the lifetime of the first bulb an...

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 λ= 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 = before integrating.]

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