Product reliability has been defined as the probability that a product will perform its intended function satisfactorily for its intended life when operating under specified conditions. The reliability function, R(x). for a product indicates the probability of the product's life exceeding x time periods. When the time until failure of a product can be adequately modeled by an exponential distribution, the product's reliability function is R(x) = e-" (Ross, Stochastic Processes, 1996).

Suppose that the time to failure (in years) of a particular product is modeled by an exponential distribution with A = .5.

a. What is the product's reliability function?

b. What is the probability that the product will perform satisfactorily for at least four years?

c. What is the probability that a particular product will survive longer than the mean life of the product?

d. If A changes, will the probability that you calculated in part c change? Explain.

e. If 10,000 units of the product are sold, approximately how many will perform satisfactorily for more than five years? About how many will fail within one year?

f. How long should the length of the warranty period be for the product if the manufacturer wants to replace no more than 5% of the units sold while under warranty?