There are 10 light bulbs in a room. The lifetimes of the light bulbs are independent and exponentially distributed with mean 0.5 years. Let X be the number of light bulbs still working in the room after 1 year.

(a) What is the probability that a randomly selected light bulb is still working after 1 year?

(b) What is the expected value of X?

(c) What is the probability that X1 ?

(d) What is the distribution of X ? Justify it.

(e) Let Y be the first time when a light bulb fails. What is the distribution of Y ?Justify it.