MA 2631 Conference 5

The exponential random variable with parameter λ > 0:

f(x) = (

λe−λx x ≥ 0

0 x < 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[e

tX] (a challenge problem).

2. Suppose that the length of a phone call in minutes is an exponential random variable with parameter

λ =

1

10 . 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 exponentially

distributed 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?

1

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