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The Algebra of Expectations For any random variable X, by definition we have: EX = μ, the mean of the population on which X is defined. VAR X = σ 2 , the variance of the population on which X is defined. Let X1, X2,…, Xn be any n random variables. combination with real number coefficients ai is (1) Then the expected value of a linear E (a1 X 1 + a 2 X 2 + ... + a n X n ) = a1 EX 1 + a 2 EX 2 + ... + a n EX n . If X1, X2,…, Xn are independent random variables, the variance of a linear combination having coefficients ai is (2) VAR (a1 X 1 + a 2 X 2 + ... + a n X n ) = a 12 VAR X 1 + a 22 VAR X 2 + ... + a n2 VAR X n . Suppose the Xi are sampling random variables, i.e., X1, X2, …, Xn are iid (independent and identically distributed). Let EX i = μ , and VAR X i = σ 2 , i = 1, 2, ..., n . Consider the linear combination having coefficients ai ≡ 1 . The expected value, using (1) n above is 1 1 1 1 1 1 E ⎛⎜ X 1 + X 2 + ... + X n ⎞⎟ = μ + μ + ... + μ = μ = EX n n n n ⎠ n ⎝n and by (2), 1 1 1 1 1 1 VAR ⎛⎜ X 1 + X 2 + ... + X n ⎞⎟ = 2 σ 2 + 2 σ 2 + ... + 2 σ 2 = σ 2 / n = VAR X n n ⎠ n ⎝n n n