二维连续概率密度

Bivariate continuous probability density

Joint Probability Density

\begin{array}{r} F (x, y) = \int_{- \infty}^{y} \int_{- \infty}^{x} f (u, v) d u d v \end{array}

则 $f (x, y)$ 为联合概率密度

F (\infty, \infty) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (u, v) d u d v = 1

\frac{\partial^{2} F (x, y)}{\partial x \partial y} = f (x, y)

\begin{array}{r} P {(X, Y) \in G} = \iint_{G} f (x, y) d x d y \end{array}

就是求在区域上的二重积分

二维随机变量 $(X, Y)$ 的密度函数为：

f (x, y) = {\begin{cases} c e^{- (3 x + 4 y)} & x > 0, y > 0 \\ 0 & 其 他 \end{cases}

求 $P {X + Y} \geq 1$ 的概率

\begin{aligned} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} f (x, y) d x d y & = c \int_{0}^{+ \infty} \int_{0}^{+ \infty} e^{- (3 x + 4 y)} d x d y \\ = c \int_{0}^{+ \infty} e^{- 3 x} d x \int_{0}^{+ \infty} e^{- 4 y} d y \\ = \frac{c}{12} = 1 \Rightarrow c = 12 \end{aligned}

\begin{aligned} P {X + Y \geq 1} & = \iint_{x + y \geq 1} f (x, y) d x d y \\ = 12 \int_{0}^{1} d x \int_{1 - x}^{+ \infty} e^{- (3 x + 4 y)} d y \\ + 12 \int_{1}^{+ \infty} d x \int_{0}^{+ \infty} e^{- (3 x + 4 y)} d y \\ = 4 e^{- 3} - 3 e^{- 4} \end{aligned}

\begin{array}{r} \iint_{x + y \geq 1} f (x, y) d x d y = \int_{0}^{1} 12 d x \int_{1 - x}^{+ \infty} e^{- (3 x + 4 y)} d y + \int_{1}^{+ \infty} 12 d x \int_{0}^{+ \infty} e^{- (3 x + 4 y)} d y \end{array}

重要

一定要注意积分的上下限！

Marginal Probability Density

由联合概率密度函数求 边缘概率密度函数：

\begin{aligned} f_{X} (x) & = \int_{- \infty}^{+ \infty} f (x, y) d y \\ f_{Y} (y) & = \int_{- \infty}^{+ \infty} f (x, y) d x \end{aligned}

\begin{aligned} F_{X} (x) & = P {X \leq x} \\ = \int_{- \infty}^{x} [\int_{- \infty}^{+ \infty} f (u, y) d y] d u \\ f_{X} (x) & = \int_{- \infty}^{+ \infty} f (x, y) d y \end{aligned}

\begin{aligned} F_{Y} (y) & = P {Y \leq y} \\ = \int_{- \infty}^{y} [\int_{- \infty}^{+ \infty} f (x, v) d y] d v \\ f_{Y} (y) & = \int_{- \infty}^{+ \infty} f (x, y) d x \end{aligned}

Tip

其实很好理解
求 $x$ 的边缘分布，则与 $y$ 无关，对 $y$ 积分得到 $x$ 的分布
求 $y$ 的边缘分布则对 $x$ 进行积分

Conditional Probability Density

条件分布函数

\begin{array}{r} F_{X ∣ Y} (x ∣ y) = \frac{f (x, y)}{f_{Y} (y)} \end{array}

\begin{array}{r} F_{Y ∣ X} (y ∣ x) = \frac{f (x, y)}{f_{X} (x)} \end{array}

对任意的 $x$ ,有 $f_{X ∣ Y} (x ∣ y) \geq 0$

\begin{array}{r} \int_{- \infty}^{+ \infty} f_{X ∣ Y} (x ∣ y) d x = 1 \end{array}

\begin{aligned} lim_{ε \to 0} P {X \leq x ∣ y - ε < Y \leq y + ε} \\ = & lim_{ε \to 0} \frac{P {X \leq x, y - ε < Y \leq y + ε}}{P {y - ε < Y \leq y + ε}} \\ = & lim_{ε \to 0} \frac{F (x, y + ε) - F (x, y - ε)}{F_{Y} (y + ε) - F_{Y} (y - ε)} \\ = & \frac{\frac{\partial F (x, y)}{\partial y}}{\frac{d}{d y} F_{Y (y)}} \end{aligned}