Expectation and Variance of an Average of Independently Identically Distributed Random Variables

4.2.5 Expectation and Variance of an Average of Independently Identically Distributed Random Variables

Let X₁,X₂,…,X_n be n independently identically distributed (i.i.d.) random variables with mean of μ and a variance of σ². Let the average of the i.i.d. random variables be denoted

1-∑n ¯X = n Xi. i=1

Then the expectation of
$[ ∑n ] ∑n E [X ¯] = E 1- Xi = 1- E [Xi ] = 1n μ = μ, n i=1 n i=1 n$ (4.2)

and the variance of
$( ) 1 ∑n 1 ∑n 1 2 σ2 V ar(X¯) = V ar -- Xi = -2- V ar(Xi ) = -2n σ = ---. n i=1 n i=1 n n$ (4.3)

Note: Since the random variables are independent the Cov(X_i,X_j) = 0 for i≠j.

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