ue to (b) If 15 passengers áre stopped by the detector, would it be unusual for none Va of these to have been stopped due to change in the pocket? Explain based 52. A on the probability of this occurring. 45. (Point binomial or Bernoulli distribution.) Assume that an ducted and that the outcome is considered to be either a success or a failure. Let p denote the probability of success. Define X to be 1 if the experiment i cess and 0 if it is a failure. X is said to have a point binomial or a Bernoulli dis. b: experiment is con. 20th 53. S suc- tribution with parameter p. (a) Argue that X is a binomial random variable with n = 1. (b) Find the density for X. (c) Find the moment generating function for X. (d) Find the mean and variance for X. (e) In DNA replication errors can occur that are chemically induced. Some of these errors are "silent" in that they do not lead to an observable mutation Growing bacteria are exposed to a chemical that has probability .14 of in- ducing an observable error. Let X be 1 if an observable mutation results. and let X be 0 otherwise. Find E[X]. daghea a i 46. A binomial random variable has mean 5 and variance 4. Find the values of n and p that characterize the distribution of this random variable. Sectio 54. S nirie Tallaiani Section 3.6 W = = =n 55. S 47. A company is manufacturing highway emergency flares. Such flares are sup- posed to burn for an average of 20 minutes. Every hour a sample of flares is collected, and their average burn time is determined. If the manufacturing process is working correctly, there is a 68% chance that the average burn time of the sample will be between 14 minutes and 26 minutes. The quality engineer in charge of the process believes that if 4 of 5 samples fall outside these bounds then this is a signal that the process might not be performing as expected. Each morning the sampling begins anew. Let X denote the number of samples drawn 56. S 57. DISCRETE DISTRIE TIATA in order to obtain the fourth sample whose average value is outside of the above bounds. Find the probability that for a given morning X = 5 and hence there seems to be a problem right away. 48. A particular pitching machine is manufactured so that it will throw the ball into the strike zone of a 6-foot batter 90% of the time. What is the average number of pitches that it will throw in order to walk a batter (that is, throw 4 pitches outside of the strike zone)? What is the probability that the fourth ball will be thrown on the seventh pitch? 49. Use the moment generating function to show that the mean of a negative bino- mial distribution with parameters r and p is r/p. 50. Use the moment generating function to show that E[X²] = (r² + rq)/p² and that Var X = rg/p² for the negative binomial distribution with parameters r and p. 51. Show that the geometric distribution is a special case of the negative binomial distribution with r = variable with parameter p using Exercises 49 and 50. Compare your answer with the results of Theorem 3.4.3. 52. A vaccine for desensitizing patients to bee stings is to be packed with three vials in each box. Each vial is checked for strength before packing. The proba- bility that a vial meets specifications is .9. Let X denote the number of vials that must be checked to fill a box. Find the density for X and its mean and variance Would you be surprised if seven or more vials have to be tested to find three that meet specifications? Explain, based on the probability of this oscu 1. Find the mean and yariance of a geometric random 53. Somo ob