兩獨立隨機變數的期望值與變異數

假設 X, Y 為獨立的隨機變數​，則：

1. \(E[X+Y]=E[X]+E[Y]\)
2. \(E[XY]=E[X]\cdot E[Y]\)
3. \(Var[X+Y]=Var[X]+Var[Y]\)

令：

\(X=\left\{x_1,x_2,\cdots,x_m\right\}\)、\(Y=\left\{y_1,y_2,\cdots,y_n\right\}\)
\(p_i=P(X=x_i)\)、\(q_j=P(Y=y_j)\)

則：

\(\begin{array}{rcl} E[X+Y]&=&\sum_{i=1}^{m}\sum_{j=1}^n (x_i+y_j)\cdot p_i q_j\\ \\ &=&\sum_{i=1}^{m}\left[p_i \cdot\sum_{j=1}^n (x_i \cdot q_j+y_j\cdot q_j)\right]\\ \\ &=&\sum_{i=1}^{m}\left[x_i p_i\cdot\sum_{j=1}^n q_j+p_i \sum_{j=1}^n y_j q_j\right]\\ \\&=&\sum_{i=1}^{m}\left[x_i p_i\cdot 1+p_i \cdot E[Y]\right]\\ \\ &=&\sum_{i=1}^{m} x_i p_i+E[Y]\cdot\sum_{i=1}^{m} p_i \\ \\ &=&E[X]+E[Y] \end{array}\)

\( \begin{array}{rcl}
E[XY]&=&\sum_{i=1}^{m}\sum_{j=1}^n x_i y_j\cdot p_i q_j\\
&=&\sum_{i=1}^{m}\left[x_i p_i \cdot\sum_{j=1}^n y_j q_j\right] \\
&=&E[X]\cdot E[Y]
\end{array}
\)

\( \begin{array}{rcl} Var[X+Y]&=&E[(X+Y)^2-E[X+Y]^2]\\ &=&E[X^2+2XY+Y^2]-(E[X]+E[Y])^2 \\ &=&E[X]\cdot E[Y] \end{array} \)