(c) Find the mean and variance for the random variable Y, the number tained when a random digit generator is activated once. Hint: The sum of the first n positive integers is n(n + 1)/2; the sum of the squares of the first 88 INTRODUCTION TO PROBABILITY AND STATISTICS n positive integers is n(n + 1)(2n + 1)/6. 35. Let the density for X be given by f(x) = ce¯* x = 1, 2, 3, . . · Jedt nsv (a) Find the value of c that makes this a density. sD A bni a அபப்டு ians 1, 2, noinctosdrbni (b) Find the moment generating function for X. (C) Use mx(t) to find E[X]. iscrowe offwel tauisvg dTgipenib (X bni ( ug EIX quedyh Section 3.5 6. LetX be binomial with parameters n = 15 and p = .2. (a) Find the expression for the density for X. (b) Find the expression for the moment generating function for X. (c) Find E[X] and Var X. (d) Find E[X], E[X²], and Var X using the moment generating function, thus verifying your answer to part (c) of this exercise. (e) Find P[X < 1] by evaluating the density directly. Compare your answer to that given in Table I of App. A.o ai Bid (f) Draw dot diagrams similar to that of Fig. 3.2 to illustrate each of these probabilities, and find the probabilities using Table I of App. A. dehsv mdbnsT STzib A CE P[X < 5] vo P[X < 5] F(9) 3e P[2 < X < 7] P[X > 3] ndomori. F(20) sd a anininido to lbino rats used to study the hormonal regulation of a metabolic pathway a iected with a drug that inhibits body synthesis of protein. The probability th pA, obem P[2 < X < 7] P[X = 10] aidpdog rat will die from the drug before the experiment is over is .2. If 10 animals eated with the drug, how many are ted