二维离散分布律

p_{i j} = P {X = x_{i}, Y = y_{i}}

性质：

\begin{array}{r} p_{i j} = P {X = x_{i}, Y = y_{j}} \geq 0 \end{array}

\begin{array}{r} \sum_{i, j}^{} p_{i j} = 1 \end{array}

联合分布律:

\begin{array}{r} P_{i j} = P {X = x_{i}, Y = y_{i}} \end{array}

边缘分布律:

\begin{aligned} p_{i \cdot} & = P {X = x_{i}} = \sum_{j} p_{i j} \\ p_{\cdot j} & = P {Y = y_{j}} = \sum_{i} p_{i j} \end{aligned}

Tip

简单理解，求 $X$ 的分布律时，就把对应的 $Y$ 累加起来（与 $Y$ 无关）
求 $Y$ 的分布律同理

\begin{array}{r} P {X = x_{i} ∣ Y = y_{i}} = \frac{P {X = x_{i}, Y = y_{i}}}{P {Y = y_{i}}} = \frac{p_{i j}}{p_{\cdot j}} \end{array}

\begin{array}{r} P {Y = y_{i} ∣ X = x_{i}} = \frac{P {X = x_{i}, Y = y_{i}}}{P {X = x_{i}}} = \frac{p_{i j}}{p_{i \cdot}} \end{array}