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hmwk6 - ISyE 2027 R D Foley Probability with Applications...

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ISyE 2027 Probability with Applications Fall 2006 R. D. Foley Homework 6 October 12, 2006 due on Thursday 1. What would be a reasonable guess as to the distribution of (a) the number of cartons of eggs purchased until you purchase a carton that contains at least one broken egg? (b) whether the next carton purchased contains three broken eggs or not? (c) the number of broken eggs in the next carton purchased? (d) the number of insects caught by a particular free range chicken between noon and 1 p.m. today? 2. Suppose X is a random variable with probability mass function Pr { X = k } = ck for k = 1 , 2 , 3 , 4. (a) Determine c . (b) Compute E[ X ]. (c) Compute the probability mass function of Y = X 2 . (d) Compute E[ Y ] from the p.m.f. of Y . (e) Recompute E[ Y ] = E[ X 2 ] using the “law of the unconscious statistician.” (f) Compute E[ X ( X - 1)] by using the answers from parts (b) and (e) and some basic properties of E[]. (g) Compute the variance of X . (h) Compute the standard deviation of X . (i) Compute the conditional probability mass function Pr { X = k | X 2 } . (j) Compute E[ X | X
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