First Midterm 2006

First Midterm 2006 - AEB 6933Mathematical Statistics for...

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AEB 6933–Mathematical Statistics for Food and Resource Economics Examination I, September 29, 2006 1. Given , f x y such that 22 31 , 10 352 4 f x y x y xy defined over , 2,2 xy . 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) 2. Given that x is distributed 0,5 U and that 2 4 y x x for 0,5 x (and 100,0 y ): a. What are the conditions required for deriving gy (the probability density function for y ) based on yx ? (10 points) b. Derive . (10 points) 3. Given the distribution function 2 3 1 2 f x x defined over the interval 0,1 x a. Compute the first four population moments of the distribution. (10 points) b. Compare these moments with the sample moments in Table 1, do you think that this sample was drawn from this population? (10 point) 4. Given that 1 2 2 3 1 ~, 1 1 5 x xN x a. Compute the conditional expectation of 2 x such that 1 0 x . (10 points) b. Compute the conditional variance of 2 x . (10 points)
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Table 1. Sample Data for Question 3
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First Midterm 2006 - AEB 6933Mathematical Statistics for...

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