An important property of certain products that are in powder or granular form is their particle size distribution. For example, refractory cements are adversely affected by too high a proportion of coarse granules, which can lead to weaknesses from poor packing. G. H. Brown (Journal of Quality Technology, July 1985) showed that the beta distribution provides an adequate model for the percentage Y of refractory cement granules in bulk form that are coarse. Suppose you are interested in controlling the proportion Y of coarse refractory cement in a lot, where Y has a beta distribution with parameters α = β = 2.
a. Find the mean and variance of Y.
b. If you will accept the lot only if less than 10% of refractory cement granules are coarse, find the probability of lot acceptance.