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2. (20 pts) Suppose you are in line at Costco and there are 100 people in...

2. (20 pts) Suppose you are in line at Costco and there are 100 people in front of you. Assume that the time it takes each person in front of you to be served follows an Exponential distribution with expected value 4 (ignore units). Let X be the total time you wait for all 100 people in front of you to be served.

(a) (2 pts) What is E[X]?

(b) (2 pts) What is Var(X)?

(c) (4 pts) What is the moment generating function of X?

(d) (4 pts) Use Markov’s inequality to bound P(X ≥ 1000).

(e) (4 pts) Use Chebyshev’s inequality to bound P(X ≥ 1000).

(f) (4 pts) Use a normal approximation to approximate P(X ≥ 1000).

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