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02_Exam - Stefan Weber Shirshendu Chatterjee ORIE Cornell...

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Stefan Weber Shirshendu Chatterjee ORIE, Cornell University Exam 2 ORIE 3500/5500: Engineering Probability and Statistics II Question 1 [20 points] (i) Define the following notions: (a) Conditional PDF of X given Y for jointly absolutely continuous random variables X and Y . (3) (b) Independence of two jointly absolutely continuous random variables X and Y . (3) (c) Moment-generating function (transform) of a random variable X . (3) (d) Moment-generating function (transform) of two random variables X and Y . (3) (ii) Please provide an answer to the following questions: (a) If X is a random variable with moment-generating function M X , and a,b R , what is the moment generating function M Y of Y = aX + b in terms of M X ? (4) (b) If X is a random variable with moment-generating function M X , how can you calcu- late the second moment of X ? (4) Question 2 [25 points] (i) Derive the moment-generating functions of the following distributions: (a) Poisson distribution with parameter λ >
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