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 226. In that regard, consider Review Exercise 3.73 on page 128. 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 1. Verify that the above is a valid density function.

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

3. What are the parameters α and β of the beta distribution illustrated here?

4. The mean of the beta distribution is . What is the mean proportion of impurities in the batch?

5. The variance of a beta distributed random variable is What is the variance of Y in this problem?