<Dd> y _̄ = 1 n ∑ i = 1 n y i . (\ displaystyle (\ overline (y)) = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) y_ (i).) </Dd> <P> Since the y are selected randomly, both y _̄ (\ displaystyle \ scriptstyle (\ overline (y))) and σ y 2 (\ displaystyle \ scriptstyle \ sigma _ (y) ^ (2)) are random variables . Their expected values can be evaluated by averaging over the ensemble of all possible samples (y) of size n from the population . For σ y 2 (\ displaystyle \ scriptstyle \ sigma _ (y) ^ (2)) this gives: </P> <Dl> <Dd> E (σ y 2) = E (1 n ∑ i = 1 n (y i − 1 n ∑ j = 1 n y j) 2) = 1 n ∑ i = 1 n E (y i 2 − 2 n y i ∑ j = 1 n y j + 1 n 2 ∑ j = 1 n y j ∑ k = 1 n y k) = 1 n ∑ i = 1 n (n − 2 n E (y i 2) − 2 n ∑ j ≠ i E (y i y j) + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E (y j y k) + 1 n 2 ∑ j = 1 n E (y j 2)) = 1 n ∑ i = 1 n (n − 2 n (σ 2 + μ 2) − 2 n (n − 1) μ 2 + 1 n 2 n (n − 1) μ 2 + 1 n (σ 2 + μ 2)) = n − 1 n σ 2 . (\ displaystyle (\ begin (aligned) E (\ sigma _ (y) ^ (2)) & = E \ left ((\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left (y_ (i) - (\ frac (1) (n)) \ sum _ (j = 1) ^ (n) y_ (j) \ right) ^ (2) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) E \ left (y_ (i) ^ (2) - (\ frac (2) (n)) y_ (i) \ sum _ (j = 1) ^ (n) y_ (j) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) y_ (j) \ sum _ (k = 1) ^ (n) y_ (k) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left ((\ frac (n - 2) (n)) E (y_ (i) ^ (2)) - (\ frac (2) (n)) \ sum _ (j \ neq i) E (y_ (i) y_ (j)) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) \ sum _ (k \ neq j) ^ (n) E (y_ (j) y_ (k)) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) E (y_ (j) ^ (2)) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left ((\ frac (n - 2) (n)) (\ sigma ^ (2) + \ mu ^ (2)) - (\ frac (2) (n)) (n - 1) \ mu ^ (2) + (\ frac (1) (n ^ (2))) n (n - 1) \ mu ^ (2) + (\ frac (1) (n)) (\ sigma ^ (2) + \ mu ^ (2)) \ right) \ \ & = (\ frac (n - 1) (n)) \ sigma ^ (2). \ end (aligned))) </Dd> </Dl> <Dd> E (σ y 2) = E (1 n ∑ i = 1 n (y i − 1 n ∑ j = 1 n y j) 2) = 1 n ∑ i = 1 n E (y i 2 − 2 n y i ∑ j = 1 n y j + 1 n 2 ∑ j = 1 n y j ∑ k = 1 n y k) = 1 n ∑ i = 1 n (n − 2 n E (y i 2) − 2 n ∑ j ≠ i E (y i y j) + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E (y j y k) + 1 n 2 ∑ j = 1 n E (y j 2)) = 1 n ∑ i = 1 n (n − 2 n (σ 2 + μ 2) − 2 n (n − 1) μ 2 + 1 n 2 n (n − 1) μ 2 + 1 n (σ 2 + μ 2)) = n − 1 n σ 2 . (\ displaystyle (\ begin (aligned) E (\ sigma _ (y) ^ (2)) & = E \ left ((\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left (y_ (i) - (\ frac (1) (n)) \ sum _ (j = 1) ^ (n) y_ (j) \ right) ^ (2) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) E \ left (y_ (i) ^ (2) - (\ frac (2) (n)) y_ (i) \ sum _ (j = 1) ^ (n) y_ (j) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) y_ (j) \ sum _ (k = 1) ^ (n) y_ (k) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left ((\ frac (n - 2) (n)) E (y_ (i) ^ (2)) - (\ frac (2) (n)) \ sum _ (j \ neq i) E (y_ (i) y_ (j)) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) \ sum _ (k \ neq j) ^ (n) E (y_ (j) y_ (k)) + (\ frac (1) (n ^ (2))) \ sum _ (j = 1) ^ (n) E (y_ (j) ^ (2)) \ right) \ \ & = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) \ left ((\ frac (n - 2) (n)) (\ sigma ^ (2) + \ mu ^ (2)) - (\ frac (2) (n)) (n - 1) \ mu ^ (2) + (\ frac (1) (n ^ (2))) n (n - 1) \ mu ^ (2) + (\ frac (1) (n)) (\ sigma ^ (2) + \ mu ^ (2)) \ right) \ \ & = (\ frac (n - 1) (n)) \ sigma ^ (2). \ end (aligned))) </Dd>

Variance of the sum is the sum of the variances