概率论笔记-随机向量及其分布

ZhongZhiy

2025-10-19 (Updated: 2026-01-18)

note › 概率论

概率论

随机向量及其分布

联合分布

$F(x, y) = P{X <= x, Y<= y}$为随机向量$(X, Y))$的分布函数, 或随机变量X, Y的联合分布函数
在坐标轴上就是在$(x, y)$下的矩形面积

性质

$0 <= F(x, y) <= 1$, 且$F(-\infty, y) = F(x, -\infty) = 0$, $F(\infty, \infty) = 1$
$F(x, y)$关于$x$和$y$单调不减
$F(x, y)$连续可微时, $f(x, y) = \frac{\partial^2}{\partial x \partial y}F(x, y)$为随机向量$(X, Y))$的联合概率密度函数
$F(x, y)$关于变量x和y右连续

二维离散型rv可以用列表表示

分布函数就是$F(x,y) = \sum_{i} \sum_{j} p_{ij} I_{x_i}(x) I_{y_j}(y)$

连续型就是$F(x, y) = \int_{-\infty}^x \int_{-\infty}^y f(u, v) du dv$

连续型的性质

$f(x, y) >= 0$
$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) dx dy = 1$
$f(x, y) = \frac{\partial^2}{\partial x \partial y}F(x, y)$
G是一个区域, 那么$P{(X, Y) \in G} = \int_{G} f(x, y) dx dy$
$P{x_1 < X <= x_2, y_1<Y<=y_2} = \int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x, y) dx dy$

均匀分布

G为平面, A为G平面的面积, 则:
$f(x, y) = \frac{1}{A}$, (x, y) \in G$

正态分布

$$f(x, y) = \frac{1}{2\pi\sigma_x\sigma_y}e^{-\frac{(x-\mu_x)^2}{2\sigma_x^2} - \frac{(y-\mu_y)^2}{2\sigma_y^2}}$, (x, y) \in \mathbb{R}^2$$

边缘分布

$$f_X(x) = \int_{-\infty}^{\infty} f(x, y) dy$$
$$f_Y(y) = \int_{-\infty}^{\infty} f(x, y) dx$$

离散型
$$f_X(x) = \sum_{y} p_{xy}$$
$$f_Y(y) = \sum_{x} p_{xy}$$

计算概率密度

$f_X(x) = \int_{-\infty}^{\infty} f(x, y) dy$
$f_Y(y) = \int_{-\infty}^{\infty} f(x, y) dx$

正态分布

$(X, Y) ~N((\mu_x, \mu_y), (\sigma_x^2, \sigma_y^2, \rho\sigma_x\sigma_y)) Rightarrow X ~N(\mu_x, \sigma_x^2), Y ~N(\mu_y, \sigma_y^2)$

example

G为$y = x^2$ and $y = x$围成的区域, $(X, Y)$服从均匀分布, 求X, Y的联合分布和边缘概率密度
$A = \int_0^1(x - x^2)dx = \frac{1}{6}$
so $f(x, y) = 6, (x, y) \in G$
when $0 <= x <= 1$, $f_X(x) = \int_{-\infty}^{infty}f(x, y)dy = \int_{-infty}^{x^2}0dx + \int_{x^2}^{x}6dx + 0 = 6(x - x ^2)$
‘

独立性

$F(x, y) = F_X(x) \cdot F_Y(y)$就称X和Y相互独立
等价于$f(x, y) = f_X(x) \cdot f_Y(y)$
正态分布X和Y互相独立的necessary and sufficient condition is $\rho = 0$

随机变量函数的分布

1. 离散型

就每个值带进去
$x_i ; -> ; f(x_i)$

连续型

X的分布密度为$f(x), y = g(x)$是单调函数, 则$Y = g(X)$概率分布为
$$
f_Y(y) = |h(y)’| f[h(y)]
$$
其中$h(x)$是$g(x)$的反函数

两个随机变量函数的分布

$(X, Y), z = g(x, y)$为连续函数, 则称$Z = g(X, Y) is (X, Y)$的函数
$$
F_Z(x) = P{Z <= x} = iint_{g(x, y)<=z} f(x, y)dxdy
$$

1. $Z = X + Y$

$$
f_Z(z) = \int_{-\infty}^{+\infty}f(x, z-x)dx \
f_Z(z) = \int_{-\infty}^{+\infty}f(z-y, y)dy
$$
若X和Y相互独立, 则有卷积公式:
$$
f_Z(z) = \int_{-\infty}^{+\infty}f_X(z-y)f_Y(y)dy \
f_Z(z) = \int_{-\infty}^{+\infty}f_X(x)f_Y(z-x)dx \
$$
若, $X~~N(\mu_x, \sigma_x^2), Y~~N(\mu_y, \sigma_y^2)$, 则$Z = X + Y~N(\mu_x + \mu_y, \sigma_x^2 + \sigma_y^2)$

若, $X~~N(\mu_x, \sigma_x^2), Y~~N(\mu_y, \sigma_y^2)$, 则$Z = k_1 X + k_2 ZN(k_1\mu_x + k_2\mu_y, K_1^2\sigma_x^2 + K_2^2\sigma_y^2)$
$Y = aX + b, YN(a\mu+b, (a\sigma)^2)$

2. $Z_1 = max{X, Y}, Z_2 = min{X, Y}$, X, Y相互独立

$$
F_{Z_1}(z) = P{Z_1 <= z} = P{X <= z, Y <= z} = F_X(z)F_Y(z)
$$
$$
f_{Z_1}(z) = F_X(z)f_Y(z) + F_Y(z)f_X(z)
$$
$$
F_{Z_2}(z) = P{Z_2 <= z} = P{X >= z, Y >= z} = (1 - F_X(z))(1 - F_Y(z))
$$
$$
f_{Z_2}(z) = -(1 - F_X(z))f_Y(z) - (1 - F_Y(z))f_X(z)
$$