[Solved] MA2631 Conference 5

The exponential random variable with parameter > 0:

1. Let X be an exponential random variable with parameter . Calculate Var[X] in two ways:
   • 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;
   • By deriving the moment generating function M_X(t) = E[e^tX] (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

• more than 10 minutes;
• 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 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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