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2027Hmwk9 - ISyE 2027 R D Foley Probability with...

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ISyE 2027 Probability with Applications Fall 2011 R. D. Foley Homework 9 November 28, 2011 due Monday 1. Let X and Y be independent, exponentially distributed random variables with parameters λ and μ , respectively. (a) Compute the c.d.f. of Z where Z = X Y . (b) Compute the p.d.f. of Z . (c) What is the name of the distribution of Z including any parameters? (d) If λ = 3 per hour and μ = 7 per hour, what is the expected value of Z in minutes? 2. Turtles A and B are racing. The length of time until A finishes the race is exponentially distributed with mean 30 minutes, and for B, exponen- tially distributed with mean 20 minutes, and for C, exponentially dis- tributed with mean 15 minutes. Assume that the three times are inde- pendent. (a) What is the probability that A takes longer than 1 hour? (b) What is the probability that A wins? (c) What is the expected time of the winning turtle in hours? (d) What is the probability that the winning time is under 10 minutes? 3. Suppose the joint p.m.f. of X and Y is P { X = i , Y = j } = c ( i + j ) for i = 0, 1, 2 , j = 0, 1 , and 0 otherwise. Determine c and then compute (a) the covariance of X
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