(X, Y)为两个随机变量, (p_X(x), p_Y(y))分别为(X, Y)的概率密度/质量函数, (p(x, y))为它们的联合概率密度.
(E(X + Y) = E(X) + E(Y))在任何条件下成立
[E(X + Y) = int_{-infty}^{{+infty}} int_{-infty}^{{+infty}} (x + y) p(x, y) dx dy
\ = int_{-infty}^{{+infty}} int_{-infty}^{{+infty}} x p(x, y) dx dy + int_{-infty}^{{+infty}} int_{-infty}^{{+infty}} y p(x, y) dx dy
\ = E(X) + E(Y)
]
不需要(X, Y)相互独立
(E(XY) = E(X)E(Y))在(X, Y)相互独立时成立
[E(XY) = int_{-infty}^{{+infty}} int_{-infty}^{{+infty}} xy p(x, y) dx dy
]
当(X, Y)相互独立时, (p(x, y) = p_X(x)p_Y(y)):
[E(XY) = int_{-infty}^{{+infty}} int_{-infty}^{{+infty}} xy p_X(x)p_Y(y) dx dy = E(X)E(Y)
]
(D(X + Y) = D(X) + D(Y))在(X, Y)相互独立时成立
[D(X + Y) = E([X + Y]^2) - E^2(X + Y) = E(X^2) + E(Y^2) + 2E(XY) - E^2(X) - E^2(Y) - 2E(X)E(Y)
]
当(X, Y)相互独立时, (2E(XY) = 2E(X)E(Y)):
[D(X + Y) = E([X + Y]^2) - E^2(X + Y) = E(X^2)- E^2(X) + E(Y^2) - E^2(Y) = D(X) + D(Y)
]