Question

Consider Exercise 3.31 on page 94. Based on extensive testing, it is determined that the time Y in years before a major repair is required for a certain washing machine is characterized by the density function

                         

Note that Y is an exponential random variable with μ = 4 years. The machine is considered a bargain if it is unlikely to require a major repair before the sixth year. What is the probability P(Y > 6)? What is the probability that a major repair is required in the first year?

Answer 1

Solution

Consider the cumulative distribution function F(y) for the exponential distribution,

Thus, the probability that the washing machine will require major repair after year six is 0.223. Of course, it will require repair before year six with probability 0.777. Thus, one might conclude the machine is not really a bargain. The probability that a major repair is necessary in the first year is

P(Y < 1) = 1 − e^{1/4} = 1 − 0.779 = 0.221.