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Unformatted text preview: ISyE 2027 Probability with Applications Fall 2011 R. D. Foley Homework 8 November 8, 2011 due Wednesday 1. Collie eye anomaly (CEA) is an autosomal simple recessive trait. Sup pose there are 5 pups in the litter and both the sire and dam are carriers of collie eye anomaly. Let X be the number of affected (homozygous recessive) pups in the litter. (a) What is the p.m.f. of X ? (b) What is the mean of X ? (c) What is the variance of X ? (d) What is P { X = i , Y = j } where Y is the number of pups in the litter that are carriers of CEA? (e) What is the mean and variance of X + Y + Z where Z is the number of pups in the litter that are clear of CEA? 2. Let Z have a Gaussian distribution with mean 0 and variance 1. Com pute and memorize P { k 6 Z 6 k } for k = 1, 2 and 3. 3. Let X have a Gaussian distribution with mean 5 and variance 9. Using only paper and pen, compute (a) P { 2 6 X 6 8 } , (b) P { 1 6 X 6 11 } , (c) P { 4 6 X 6 14 } , (d) P { X > 2 } , (e) P { 2 6 X 6 14 } , (f) P { X 5  > 6 } and compare this answer with the bound obtained from Chebyshevs inequality. 4. Let X have a Gaussian distribution with mean 3 and variance 25, and let Z have a standard normal distribution. Using Tables 6.1 and 6.2 in the back of our textbook, compute (a) P { X 6 12.8 } , (b) P { X 3  > 9.8 } , (c) P { Z > .87 } , (d) P { Z 6 .87 } . (e) P { X > 1  X > 2 } , 1 5. Let X have a binomial distribution with parameters n = 15 and p = 1 / 5. Let Y have a normal distribution with same mean and variance as X . (a) Using the continuity correction, give the event involving Y that would be used to approximate each of the following events involv ing X : { X 6 3 } , { X < 3 } , { X = 3 } , { 2 < X 6 4 } ....
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 Fall '08
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