Solved: In the text of this section, we noted the relationship between the distribution

Chapter 4, Problem 4.135

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In the text of this section, we noted the relationship between the distribution function of a beta-distributed random variable and sums of binomial probabilities. Specifically, if Y has a beta distribution with positive integer values for and and 0 < y < 1, F(y) = " y 0 t1(1 t)1 B(, ) dt = n i= n i yi (1 y) ni , where n = + 1. a If Y has a beta distribution with = 4 and = 7, use the appropriate binomial tables to find P(Y .7) = F(.7). b If Y has a beta distribution with = 12 and = 14, use the appropriate binomial tables to find P(Y .6) = F(.6). c Applet Exercise Use the applet Beta Probabilities and Quantiles to find the probabilities in parts (a) and (b).c If k is an integer between 1 and n 1, the same argument used in part (b) yields that P(Y2 k) = k i=0 n i (p2) i (1 p2) ni = " 1 p2 t k (1 t)nk1 B(k + 1, n k) dt. Show that, if k is any integer between 1 and n 1, P(Y1 k) > P(Y2 k). Interpret this result.