First Midterm 2007

# First Midterm 2007 - Examination I October 1 2006 I Given f...

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Examination I October 1, 2006 I. Given , f x y such that 2 2 5 7 1 3 1 1 , 24 24 6 8 24 6 f x y x x y xy y Defined over , 0,2 x y . A. Is this a valid probability density function? (10 Points) B. Derive the marginal distribution of y . (10 Points) C. Derive the conditional distribution of x given y . (10 Points) D. Are x and y independent? (10 Points). II. Given that x is distributed 0,8 U and that 1 3 y x A. What are the conditions for deriving g y (the probability density function for y ) based on   y x ? (10 Points) B. Derive the distribution g y . (10 Points). III. Assume 2 9 y f y where 0,3 y . A. Derive the moment generating function for y . (10 Points) B. Compute the first three moments for this distribution. (10 Points) C. Using the random variables in Table 1, calculate the first three sample moments. Do you think that these random variables are from f y ? (10 Points) IV. Given 1 2 4 2 1 ~ , 3 1 5 x N x

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