471_II_solutions - (2(a P XY> 3 = Z 3 1 Z 3 3/y 4 xy 81 dxdy(b f X x = Z 3 4 xy 81 dy = 2 x 9(c The random variables are independent since the

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Prelim II 2005 Solutions (1) For the mean, see Example 1.7 in Section 4.1 or use the fact that a gamma( n,λ ) is a sum of n independent exponential( λ ) random variables: E (gamma(7 )) = 7 X i =1 1 λ = 7 λ . For the variance, see Example 4.2 in Section 4.2. Note that when you integrate by parts, you must evaluate a limit which is easy to evaluate using either l’Hospital’s Rule, or the fact that e x grows faster than any polynomial.
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Unformatted text preview: (2) (a) P ( XY > 3) = Z 3 1 Z 3 3 /y 4 xy 81 dxdy (b) f X ( x ) = Z 3 4 xy 81 dy = 2 x 9 (c) The random variables are independent since the joint density is a product of the marginal densities (see Equation 5.5 on page 125). (3) When 0 < z < 3, f X + Y ( z ) = Z z 2 x 9 2( z-x ) 9 dx. When 3 < z < 6, f X + Y ( z ) = Z 3 z-3 2 x 9 2( z-x ) 9 dx....
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).

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