The exponential random variable with parameter λ > 0:
1. Let X be an exponential random variable with parameter λ. Calculate Var[X] in two ways:
(a) By looking up E[X] in the lecture notes, calculating E[X2] directly using the definition of expectation, and the formula Var[X] = E[X2] − (E[X])2;
(b) By deriving the moment generating function MX(t) = E[etX] (a challenge problem).
2. Suppose that the length of a phone call in minutes is an exponential random variable with parameter
. If someone arrives immediately ahead of you at public telephone booth, find the probability you will have to wait
(a) more than 10 minutes;
(b) not more than one standard deviation away from the mean.
3. We say that a nonnegative random variable X is memoryless if
P[X > s + t|X > t] = P[X > s] for all s,t ≥ 0.
Show that an exponential random variable X with parameter λ is memoryless.
4. Suppose that the number of miles that a car can run before its battery wears out is exponentiallydistributed with an average value of 10,000 miles. If a person desires to take a 5,000 mile trip, what is the probability that he or she will be able to complete the trip without having to replace the car battery?
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