Notes on Variance , Covariance , and Summation Operator By Hiro Kasahara

Notes on Variance, Covariance, and Summation Operator By Hiro Kasahara Properties of Summation Operator For a sequence of the values {x1, x2, ..., xn}, we write the sum of x1, x2, ..., xn−1, and xn using the summation operator as x1 + x2 + ... + xn = n ∑ i=1 xi. (1) Given a constant c, n ∑ i=1 cxi = cx1 + cx2 + ... + cxn = c× (x1 + x2 + ... + xn) = c n ∑ i=1 xi. (2) • For example, consider the case that n = 2 with the values of {x1, x2} given by x1 = 0 and x2 = 1. Suppose that c = 4. Then, ∑2 i=1 4× xi = 4× 0 + 4× 1 = 4× (0 + 1) = 4 ∑2 i=1 xi. • In the special case of x1 = x2 = ... = xn = 1, we have ∑n i=1 cxi = ∑n i=1 c× 1 = c× ∑n i=1 1 = c× (1 + 1 + ... + 1) = nc. Consider another sequence {y1, y2, ..., ym} in addition to {x1, x2, ..., xn}. Then, we may consider double summations over possible values of x’s and y’s. For example, consider the case of n = m = 2. Then, ∑2 i=1 ∑2 j=1 xiyj is equal to x1y1 + x1y2 + x2y1 + x2y2 because x1y1 + x1y2 + x2y1 + x2y2 = x1(y1 + y2) + x2(y1 + y2) (by factorization) = 2 ∑ i=1 xi(y1 + y2) (by def. of the summation operator by setting c = (y1 + y2) in (2) )