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