Question

Suppose hard drive A has a lifetime that is exponentially distributed with mean of 6 years and hard drive B has a lifetime that is exponentially distributed with a mean of 2 years. What is the probability that drive B lasts at least 3 times longer than drive A?

Answer 1

Let X denote the lifetime of first hard drive and Y denote the lifetime of hard drive B. We can assume that the lifetime of each hard drive are independent.

Given X has exponential distribution with mean 6 and Y has exponential distribution with mean 2.

We want to find

This probability is given by

since X and Y are independent.

where deonte the probability density function of X and denote the probability density function of Y.

Now the density functions are given by

and .

Now,

Hence the required probability

So the probability that drive B lasts at least 3 times longer than drive A is