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IE111_F09_HW06_soln - IE 111 Fall 2009 Homework#6 Solutions...

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IE 111 Fall 2009 Homework #6 Solutions Question 1. To check on the security screeners at an airport, we put 4 fake bombs into a set of 52 baggage pieces. Suppose that the screeners actually just pick 6 pieces randomly (without replacement) to inspect. Let X be the number of our fake bombs that they find. Find V(X) V[X] = 0* P X (x=0) + 1* P X (x=1)+ 4* P X (x=2)+ 9* P X (x=3)+ 16* P X (x=4) – E[X] 2 = 0.384 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? 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? 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. The total amount of snow during a winter in the Lehigh valley is a random variable with mean 40 inches and variance 100 inches. My son is wishing for 100 or more inches of snow this winter. What can you tell him about the probability of his wish? μ = 40 σ = 10 If we choose k=6 then Chebyshev’s inequality becomes: P( μ -k σ < X < μ +k σ ) 1 - (1/k 2 ) P(40 -6*10 < X < 40+6*10) 1 - (1/6 2 ) P(-20 < X < 100) 0.97222 So his chances are less than 0.028 1
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Question 4.
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