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

Question:

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 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? 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?
Distribution
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