測試數學式

$$ \begin{cases} x^2& \text{ ,if } 2\leq x\leq 3 \\ 2x-3& \text{ ,if } x= 4 \end{cases} $$ $$ \begin{array}{rcl} E[X+Y]&=&\sum_{i=1}^{m}\sum_{j=1}^n (x_i+y_j)\cdot p_i q_j\\ \\ &=&\sum_{i=1}^{m}\left[p_i \cdot\sum_{j=1}^n (x_i \cdot q_j+y_j\cdot q_j)\right]\\ \\ &=&\sum_{i=1}^{m}\left[x_i p_i\cdot\sum_{j=1}^n q_j+p_i \sum_{j=1}^n y_j q_j\right]\\ \\ &=&\sum_{i=1}^{m}\left[x_i p_i\cdot 1+p_i \cdot E[Y]\right]\\ \\ &=&\sum_{i=1}^{m} x_i p_i+E[Y]\cdot\sum_{i=1}^{m} p_i \\ \\ &=&E[X]+E[Y] \end{array} $$