離散型確率変数(discrete random variable) / 期待値(expected value)

$$\begin{array}{rcl} 母集団(population)の確率変数&:&X,Y\\ 確率質量凾数(probability\,mass\,function,\,PMF)&:&f_X(x)=P(X=x_i)\\ &&\displaystyle\sum_{i=1}^{\infty} f_X(x_i)=1\\ 累積分布凾数(cumulative\,distribution\,function,\,CDF)&:&\displaystyle F_X(x)=\sum_{i:x_i \leq x} f_X(x_i)\\ 同時確率質量凾数(joint\,probability\,mass\,function)&:&f_X(x, y)=P(X=x_i,\,Y=y_j)\\ &&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} f_{XY}(x_i, y_j)=1\\ 周辺確率質量凾数(marginal\, probability\, mass\, function)&:&\displaystyle f_X(x)=\sum_{j=1}^{\infty} f_{XY}(x, y_i)\\ \end{array}$$ $$\begin{array}{rcl} E[g(X)]&\equiv&\displaystyle\sum_{i=1}^{\infty} g(x_i) f_X(x_i) \\ E[X]&=&\displaystyle\sum_{i=1}^{\infty} (x_i) f_X(x_i) \\ E[cX] &=&\displaystyle\sum_{i=1}^{\infty} (c x_i) f_X(x_i) \,\dotso\,cは定数\\ &=&c\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i) \\ &=&c E[X] \\ E[X \pm t]&=&\displaystyle\sum_{i=1}^{\infty} (x_i \pm t) f_X(x_i) \,\dotso\,tは定数\\ &=&\displaystyle\sum_{i=1}^{\infty} (x_i f_X(x_i) \pm t f_X(x_i)) \\ &=&\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i) \pm \displaystyle\sum_{i=1}^{\infty} t f_X(x_i) \\ &=&\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i) \pm t \displaystyle\sum_{i=1}^{\infty} f_X(x_i) \\ &=&E[X] \pm t\\ E[X \pm Y]&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (x_i \pm y_j) f_{XY}(x_i, y_j) \\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (x_i f_{XY}(x_i, y_j) \pm y_j f_{XY}(x_i, y_j)) \\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} x_i f_{XY}(x_i, y_j) \pm \displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} y_j f_{XY}(x_i, y_j)\\ &=&\displaystyle\sum_{i=1}^{\infty} x_i \sum_{j=1}^{\infty} f_{XY}(x_i, y_j) \pm \displaystyle\sum_{j=1}^{\infty} y_j \sum_{i=1}^{\infty}f_{XY}(x_i, y_j)\\ &=&\displaystyle\sum_{i=1}^{\infty} x_i f_{X}(x_i) \pm \displaystyle\sum_{j=1}^{\infty} y_j f_{Y}(y_j)\,\dotso\,周辺確率質量凾数を適用\\ &=& E[X] \pm E[Y] \\ \end{array}$$

X,Yが独立の場合

$$\begin{array}{rcl} E[XY]&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i y_j)f_{XY}(x_i, y_j)\\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}x_iy_jf_{X}(x_i)f_{Y}(y_i) \,\dotso\,X,Yが独立\,f_{XY}(x, y)=f_{X}(x)f_{Y}(y)\\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}x_if_{X}(x_i)y_jf_{Y}(y_i)\\ &=&\displaystyle\left(\sum_{i=1}^{\infty}x_if_{X}(x_i)\right)\left(\sum_{j=1}^{\infty} y_jf_{Y}(y_i)\right) \,\dotso\,\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_ib_j=\sum_{i=1}^{\infty}a_i\sum_{j=1}^{\infty}b_j\\ &=&E[X]E[Y] \end{array}$$