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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
,λ
)) =
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( zx ) 9 dx. When 3 < z < 6, f X + Y ( z ) = Z 3 z3 2 x 9 2( zx ) 9 dx....
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).
 '08
 PROTSAK
 Variance

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