First Midterm 2008

First Midterm 2008 - Examination I October 1 2008 1 Given f...

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Examination I October 1, 2008 1. Given ( ) , f xy () 222 36 9 9 9 , 64 64 64 64 2 x y xy =− + Defined over [ ] ,2 , ∈− 2 . a. Is this a valid distribution function? (5 Points) b. Derive the marginal distribution functions for x and y . (10 Points) c. Are these variables independent? (5 Points) d. Compute the conditional distribution function for x given 0 y = . (5 Points). 2. Given 2 3 125 x fx = for [ ] 0,5 x and 43 yx = : a. What are the conditions for deriving ( ) gy based on ( ) y φ = x ? (5 Points). b. Derive the distribution function ( ) including the bounds of the new random variable. (10 Points) 3. Using the distribution , x f xe λ λλ = for [ ) 0, x : a. Derive the moment generating function for x . (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 ( ) 3 , 10 ? (10 Points) 4. Given 36 2 , 22 3 xN ⎛− ⎛⎞ ⎜⎟ −− ⎝⎠ a. Compute the conditional expectation of 2 x such that 1 2
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First Midterm 2008 - Examination I October 1 2008 1 Given f...

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