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Stat HW6

# Stat HW6 - IE 111 Spring 2009 Homework#6 Due Wednesday...

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IE 111 Spring 2009 Homework #6 Due Wednesday, March 18 Question 1 Suppose you go shopping at a convenience store, pick your items, and then join the line for the cash register. There are 6 people in the line ahead of you, plus one person who is currently being served (there is only one clerk on duty). Having nothing better to do, you estimate that the time it takes to ring up one person’s items (measured in seconds) has a Geometric distribution with parameter p=1/30. a) How long do you expect to wait until the clerk starts serving you? Each customers service time is Geo~(p=1/30) in seconds. Since the Geometric is memoryless, it doesn’t matter that the first customer is already being served. There are 7 people in front of you. This means there should be 7 successes before the server starts serving you. This defines a new distribution, Y ~ Negative binomial (p=1/30, r=7) E(Y)= r/p = 7*30 =210 seconds b) You notice that the person being served right now was being served when you first entered the store 5 minutes ago. That is, it is taking a long time to serve them. How does this change your estimate for part (a)? It does not change anything because of lack of memory property of the distribution. E(Y) is still 210 seconds. c) How long do you expect to wait until you’re finished being served? Negative binomial (r=8, p=1/30) ,E(Z)= 8/1/30=240 seconds. d) What is the standard deviation of the time until you’re finished? Express it in seconds, and also, express it as a percent of the mean. V(Z) = r*(1-p)/p 2 =8*29/30*30*30= 6960, σ = Sqrt (V(Z)) = 83.43 σ / μ *100 = 83.43/240 *100 = 34.76% Question 2 The “coefficient of variation” is defined to be the standard deviation divided by the mean. It is what we use when we say something like “cars weigh 2000 pounds, plus or minus 15%”—the 15% is the coefficient of variation. Recall that the standard deviation is the square-root of the variance. a) Let V be a Binomial random variable with n=100,p=0.40; what is the coefficient of variation?

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CF = sqrt(V[X])/E[X] = sqrt(np(1-p))/np = 0.122 b) Let Y be a Pascal random variable with r=2, p=1/20. What is its coefficient of variation? Compare to X and Y from parts (a) and (b), above. CF = sqrt(V[X])/E[X] = sqrt(r(1-p)/p 2 )/(r/p) = sqrt((1-p)/r) = 0.689 c) Let Z be a Pascal random variable with r=4, p=1/10. What is its coefficient of variation? CF = sqrt(V[X])/E[X] = sqrt(r(1-p)/p 2 )/(r/p) = sqrt((1-p)/r) = 0.474 Question 3 Let the random variable X be the demand for newspapers at a news stand. X is distributed as follows: x 10 11 12 13 14 15 16 17 18 19 20 ------------------------------------------------------------------------------------ P X (x) .05 .05 .1 .1 .1 .2 .1 .1 .1 .05 .05 At the start of the day the newsboy buys papers for 20 cents a piece. He sells papers for 35 cents a piece. If there are any papers left over at the end of the day, he may sell the papers to the recycling company for 5 cents a piece. How many papers should the newsboy buy at the start of
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Stat HW6 - IE 111 Spring 2009 Homework#6 Due Wednesday...

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