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www.downloadslide.com Exercises 137 situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem? 3.133 Assume that the tunnel in Exercise 3.132 is observed during ten two-minute intervals, thus giving ten independent observations Y 1 , Y 2 ,..., Y 10 , on the Poisson random variable. Find the probability that Y > 3 during at least one of the ten two-minute intervals. 3.134 Consider a binomial experiment for n = 20, p = .05. Use Table 1, Appendix 3, to calculate the binomial probabilities for Y = 0, 1, 2, 3, and 4. Calculate the same probabilities by using the Poisson approximation **with** λ = np. Compare. 3.135 A salesperson has found that the probability of a sale on a single contact is approximately .03. If the salesperson contacts 100 prospects, what is the approximate probability of making at least one sale? 3.136 Increased research and discussion have focused on the number of illnesses involving the organism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic outbreaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. a b If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state’s mean E.coli rate has changed? Explain. 3.137 The probability that a mouse inoculated **with** a serum will contract a certain disease is .2. Using the Poisson approximation, find the probability that at most 3 of 30 inoculated mice will contract the disease. 3.138 Let Y have a Poisson distribution **with** mean λ. Find E[Y (Y − 1)] and then use this to show that V (Y ) = λ. 3.139 In the daily production of a certain kind of rope, the number of defects per foot Y is assumed to have a Poisson distribution **with** mean λ = 2. The profit per foot when the rope is sold is given by X, where X = 50 − 2Y − Y 2 . Find the expected profit per foot. ∗ 3.140 A store owner has overstocked a certain item and decides to use the following promotion to decrease the supply. The item has a marked price of $100. For each customer purchasing the item during a particular day, the owner will reduce the price by a factor of one-half. Thus, the first customer will pay $50 for the item, the second will pay $25, and so on. Suppose that the number of customers who purchase the item during the day has a Poisson distribution **with** mean 2. Find the expected cost of the item at the end of the day. [Hint: The cost at the end of the day is 100(1/2) Y , where Y is the number of customers who have purchased the item.] 3.141 A food manufacturer uses an extruder (a machine that produces bite-size cookies and snack food) that yields revenue for the firm at a rate of $200 per hour when in operation. However, the extruder breaks down an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is R = 1600 − 50Y 2 . Find the expected daily revenue for the extruder. ∗ 3.142 Let p(y) denote the probability function associated **with** a Poisson random variable **with** mean λ. a Show that the ratio of successive probabilities satisfies b For which values of y is p(y) >p(y − 1)? p(y) p(y − 1) = λ , for y = 1, 2,.... y

www.downloadslide.com 138 Chapter 3 Discrete Random Variables and Their Probability Distributions c Notice that the result in part (a) implies that Poisson probabilities increase for awhile as y increases and decrease thereafter. Show that p(y) maximized when y = the greatest integer less than or equal to λ. 3.143 Refer to Exercise 3.142 (c). If the number of phone calls to the fire department, Y , in a day has a Poisson distribution **with** mean 5.3, what is the most likely number of phone calls to the fire department on any day? 3.144 Refer to Exercises 3.142 and 3.143. If the number of phone calls to the fire department, Y ,in a day has a Poisson distribution **with** mean 6, show that p(5) = p(6) so that 5 and 6 are the two most likely values for Y . 3.9 Moments and Moment-Generating Functions The parameters µ and σ are meaningful numerical descriptive measures that locate the center and describe the spread associated **with** the values of a random variable Y . They do not, however, provide a unique characterization of the distribution of Y . Many different distributions possess the same means and standard deviations. We now consider a set of numerical descriptive measures that (at least under certain conditions) uniquely determine p(y). DEFINITION 3.12 The kth moment of a random variable Y taken about the origin is defined to be E(Y k ) and is denoted by µ ′ k . Notice in particular that the first moment about the origin, is E(Y ) = µ ′ 1 = µ and that µ ′ 2 = E(Y 2 ) is employed in Theorem 3.6 for finding σ 2 . Another useful moment of a random variable is one taken about its mean. DEFINITION 3.13 The kth moment of a random variable Y taken about its mean, or the kth central moment of Y , is defined to be E[(Y − µ) k ] and is denoted by µ k . In particular, σ 2 = µ 2 . Let us concentrate on moments µ ′ k about the origin where k = 1, 2, 3,.... Suppose that two random variables Y and Z possess finite moments **with** µ ′ 1Y = µ ′ 1Z ,µ′ 2Y = µ′ 2Z ,..., µ′ jY = µ′ jZ , where j can assume any integer value. That is, the two random variables possess identical corresponding moments about the origin. Under some fairly general conditions, it can be shown that Y and Z have identical probability distributions. Thus, a major use of moments is to approximate the probability distribution of a random variable (usually an estimator or a decision maker). Consequently, the moments µ ′ k , where k = 1, 2, 3,..., are primarily of theoretical value for k > 3. Yet another interesting expectation is the moment-generating function for a random variable, which, figuratively speaking, packages all the moments for a random variable

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