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 following density function. Complete parts (a) through (e) below. f(y) = 1 4 (1 - y), 0sys1, elsewhere (a) Verify that the given function is a valid density function. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. O A. The given function is a valid density function because it is at least 0 everywhere. O B. The given function is a valid density function because it is at least 0 everywhere and because e [ 4(1- y)3 dy =] O C. The given function is a valid density function because (b) A batch is considered not sellable and then not acceptable if the percentage of impurities exceeds 65%. What is the probability that a batch is considered not acceptable (i.e., Y > 0.65)? The probability is. (Round to four decimal places as needed.) (c) What are the parameters a and B of the beta distribution illustrated here? a = (Type an integer or a decimal. Do not round.) B =0 (Type an integer or a decimal. Do not round.) (d) The mean of the beta distribution is a a +B . What is the mean proportion of impurities in the batch? H = (Round to four decimal places as needed.) aB (e) The variance of a beta distributed random variable is of = . What is the variance of Y in this problem? (a + B)-(a + B+1) (Round to six decimal places as needed.)