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Unformatted text preview: ndom variable, Sr , with parameters r and
p, arises as the sum of r independent geometrically distributed random variables with parameter
r
p. Each such geometric random variable has mean 1/p and variance (1 − p)/p2 . So, E [Sr ] = p and
Var(Sr ) = r(1−p) .
p2 .
(c) Likewise, as seen in Section 3.5, a gamma random variable, Tr , with parameters r and λ,
arises as the sum of r independent exponentially distributed random variables with parameter λ.
An exponentially distributed random variable with parameter λ has mean 1/λ and variance 1/λ2 .
r
r
Therefore, E [Tr ] = λ and Var(Tr ) = λ2 . 4.8. MOMENTS OF JOINTLY DISTRIBUTED RANDOM VARIABLES 153 Example 4.8.2 Simplify the following expressions:
(a) Cov(8X + 3, 5Y − 2), (b) Cov(10X − 5, −3X + 15), (c) Cov(X+2,10X3Y), (d) ρ10X,Y +4 .
Solution (a) Cov(8X + 3, 5Y − 2) = Cov(8X, 5Y ) = 40Cov(X, Y ).
(b) Cov(10X − 5, −3X + 15) = Cov(10X, −3X ) = −30Cov(X, X ) = −30Var(X ).
(c) Cov(X+2,10X3Y)=Cov(X,10X3Y)=10Cov(X,X)3Cov(X,Y)=10Var(X)3Cov(X,Y).
(d) Since Cov(10X, Y +4) = 10Co...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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