\(X, Y\)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, \(2E(XY) = 2E(X)E(Y)\):
\[D(X + Y) = E([X + Y]^2) - E^2(X + Y) = E(X^2)- E^2(X) + E(Y^2) - E^2(Y) = D(X) + D(Y) \]