The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion, as in the practical scenario illustrated in Exercise 6.50 on page 206. In that regard, consider Review Exercise 3.73 on page 108. Impurities in batches of product of a chemical process reflect a serious problem. It is known that the proportion of impurities Y in a batch has the density function f(y) = 10(1 − y)

9, 0 ≤ y ≤ 1, 0, elsewhere.

(a) Verify that the above is a valid density function.

(b) What is the probability that a batch is considered not acceptable (i.e., Y > 0.6)?

(c) What are the parameters α and β of the beta distribution illustrated here?

(d) The mean of the beta distribution is α

α+β . What is the mean proportion of impurities in the batch?

(e) The variance of a beta distributed random variable is

σ2 = αβ

(α + β)2(α + β + 1).

What is the variance of Y in this problem?