Example 492 suppose a fair die is rolled 1000 times

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Unformatted text preview: Var(Y ), and this calculation extends to three or more random variables For example, consider the sum Sn = X1 + · · · + Xn , such that X1 , · · · , Xn are uncorrelated (so Cov(Xi , Xj ) = 0 if i = j ) with E [Xi ] = µ and Var(Xi ) = σ 2 for 1 ≤ i ≤ n. Then E [Sn ] = nµ (4.21) and n Var(Sn ) = Cov(Sn , Sn ) = Cov Xi , i=1 n n Xj j =1 n = Cov(Xi , Xj ) i=1 j =1 n = Cov(Xi , Xi ) + i=1 n Cov(Xi , Xj ) i,j :i=j Var(Xi ) + 0 = nσ 2 . = (4.22) i=1 Therefore, the standardized version of Sn is the random variable Sn −nµ √ . nσ 2 Example 4.8.1 Identify the mean and variance of (a) a binomial random variable with parameters n and p, (b) a negative binomial random variable with parameters r and p, and (c) a gamma random variable with parameters r and λ. Solution: (a) A binomial random variable with parameters n and p has the form S = X1 + . . . + Xn , where X1 , . . . , Xn are independent Bernoulli random variables with parameter p. So E [Xi ] = p for each i, Var(Xi ) = p(1 − p), and, since the Xi ’s are independent, they are uncorrelated. Thus, by (4.21) and (4.22), E [S ] = np and Var(S ) = np(1 − p). (b) Similarly, as seen in Section 2.6, a negative binomial ra...
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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.

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