Consider Exercise 6.50 on page 206. The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion in the practical scenario illustrated in the example. In that regard, consider Review Exercise 3.75. Impurities in the batch 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

(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 a and 0 of the beta distribution illustrated here?

(d) The mean of the beta distribution is ?/? + ?. What is the mean proportion of impurities in the batch?

(e) The variance of a beta distributed random variable is what is the variance of Y in this problem?

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S10(1 - y)", 0sy s1. 10, fiy) %3D elsewhere. Part (e) aß (a + B) (a + B+1)