Mulit-dimension variables and distribution

1 2-dimensional discrete random variables

$$P(X=x_i,Y=y_j)=p_{ij} \quad i,j=1,2,\ldots$$

2 2-dimensional continuous random variables

If the 2-dimensional variables $X,Y$'s ditributional function is $F(x,y)$, and statisfiy the equation: $$F(x,y)=\int_{-\infty}^{y}\int_{-\infty}^{x}f(u,v)dudv$$ for any $x,y$。

2.1 Properties

• $f(x,y)\ge 0$
• $\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,y)dxdy=1$
• if $D$ is a region in the $XOY$ plane, then $P((X,Y)\in D)={\iint_D}f(x,y)dxdy$
• $\frac{\partial{F(x,y)}^{2}}{\partial {x}\partial{y}}=f(x,y)$

2.2 2-dimensional uniform distribution

$$ f(x,y)= \begin{cases} \frac{1}{S_D}, & (x,y) \in D \\ 0, others \end{cases} $$

2.3 2-dimensional uniform distribution

$$ f(x,y)=\frac{1}{2\pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}}e^{\frac{-1}{2(1-\rho^2)}\big[\frac{(x-\mu_1)^2}{\sigma_1^2}-2\rho\frac{(x-\mu_1)(y-\mu_2)}{\sigma_1\sigma_2}+\frac{(y-\mu_2)^2}{\sigma_2^2}\big]} $$

3 Marginal Distribution Law

$F_x(x)=P(X\le x)=P(X\le x,y<+\infty)=F(x,+\infty)$

$F_y(y)=P(Y\le y)=P(x\le +\infty,Y<y)=F(+\infty,y)$

3.1 Discrete Marginal Distribution Law

$p_{i\cdot}=P(X=x_i)=\sum_{j=1}^{+\infty}P_{ij}$

$p_{\cdot j}=P(Y=y_j)=\sum_{i=1}^{+\infty}P_{ij}$

3.2 Marginal Probability density function

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

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

4 2-dimensional conditional distribution

4.1 Discrete conditional distribution law

$P(X=x_i | Y=y_j)=\frac{P(X=x_i,Y=y_j)}{P(Y=y_j)}=\frac{p_{ij}}{p_{\cdot j}}$

$P(Y=y_j | X=x_i)=\frac{P(X=x_i,Y=y_j)}{P(X=x_i)}=\frac{p_{ij}}{p_{i \cdot}}$

4.2 conditional probability density

$f_{X|Y}(x|y)=\frac{f(x,y)}{f_y{(y)}}$

$f_{Y|X}(y|x)=\frac{f(x,y)}{f_x{(x)}}$

4.3 conditional probability distribution

$F_{X|Y}(x|y)=P(X\le x | Y=y)=\int_{-\infty}^{x}\frac{f(x,y)}{f_Y(y)}dx$

$F_{Y|X}(y|x)=P(Y\le y | X=x)=\int_{-\infty}^{y}\frac{f(x,y)}{f_X(x)}dy$