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274 Chapter 4 Special Distributions Questions 4.6.1. An Arctic weather station has three electronic wind gauges. Only one is used at any given time. The lifetime of each gauge is exponentially distributed with a mean of one thousand hours. What is the pdf of Y , the random variable measuring the time until the last gauge wears out? 4.6.2. A service contact on a new university computer system provides twenty-four free repair calls from a technician. Suppose the technician is required, on the average, three times a month. What is the average time it will take for the service contract to be fulï¬lled? 4.6.3. Suppose a set of measurements Y1 , Y2 , . . . , Y100 is taken from a gamma pdf for which E(Y ) = 1.5 and Var(Y ) = 0.75. How many Yi âs would you expect to ï¬nd in the interval [1.0, 2.5]? 4.6.4. Demonstrate that Î» plays the role of a scale parameter by showing that if Y is gamma with parameters r and Î», then Î»Y is gamma with parameters r and 1. 4.6.5. Show that a gamma pdf has the unique mode Î»r r â1 ; Î» that is, show that the function f Y (y) = (r ) y r â1 eâÎ»y takes its and at no other point. maximum value at ymode = r â1 Î» 1 â 4.6.6. Prove that 2 = Ï. [Hint: Consider E(Z 2 ), where Z is a standard normal random variable.] â 4.6.7. Show that 72 = 158 Ï. 4.6.8. If the random variable Y has the gamma pdf with integer parameter r and arbitrary Î» > 0, show that E(Y m ) = [Hint: Use the fact that a positive integer.] &â 0 (m + r â 1)! (r â 1)!Î»m y r â1 eây dy = (r â 1)! when r is 4.6.9. Differentiate the gamma moment-generating function to verify the formulas for E(Y ) and Var(Y ) given in Theorem 4.6.3. 4.6.10. Differentiate the gamma moment-generating function to show that the formula for E(Y m ) given in Question 4.6.8 holds for arbitrary r > 0. 4.7 Taking a Second Look at Statistics (Monte Carlo Simulations) Calculating probabilities associated with (1) single random variables and (2) functions of sets of random variables has been the overarching theme of Chapters 3 and 4. Facilitating those computations has been a variety of transformations, summation properties, and mathematical relationships linking one pdf with another. Collectively, these results are enormously effective. Sometimes, though, the intrinsic complexity of a random variable overwhelms our ability to model its probabilistic behavior in any formal or precise way. An alternative in those situations is to use a computer to draw random samples from one or more distributions that model portions of the random variableâs behavior. If a large enough number of such samples is generated, a histogram (or density-scaled histogram) can be constructed that will accurately reï¬ect the random variableâs true (but unknown) distribution. Sampling âexperimentsâ of this sort are known as Monte Carlo studies. Real-life situations where a Monte Carlo analysis could be helpful are not difï¬cult to imagine. Suppose, for instance, you just bought a state-of-the-art, highdeï¬nition, plasma screen television. In addition to the pricey initial cost, an optional warranty is available that covers all repairs made during the ï¬rst two years. According to an independent laboratoryâs reliability study, this particular television is likely to require 0.75 service call per year, on the average. Moreover, the costs of service calls are expected to be normally distributed with a mean (Î¼) of $100 and a standard deviation (Ï ) of $20. If the warranty sells for $200, should you buy it?