Chapter 3, Distribution of Two-Dimensional Random Variables
Joint Distribution Function of Two-Dimensional Random Variables
- Joint distribution: F(x,y) = P(X<=x,Y<=y), where ',' means 'and'
- Marginal distribution:
F_X(x) = F(x,+ \infty)
F_Y(y) = F(+ \infty,y)
Two-dimensional discrete random variable
- Joint distribution columns:
P_{ij} = P(X=x_i,Y=y_j)
- Marginal Distribution Column: Pivot Table
- Independence:
P(X=x_i,Y=y_j) = P(X=x_i)P(Y=y_j)
Two-dimensional continuous random variable
- Joint distribution:
F(x,y) = \int_{- \infty}^{x}\int_{- \infty}^{y}f(u,v)dudv
- Marginal density function:
f_X(x) =F(x,+ \infty ) ' = (\int_{- \infty}^{+ \infty}\int_{- \infty}^{x}f(x,y)dxdy)' = \int_{- \infty}^{+ \infty}f(x,y)dy
f_Y(y) = \int_{- \infty}^{+ \infty}f(x,y)dx
- Independence:
f(x,y) = f_X(x) f_Y(y)
Distribution of a function of a two-dimensional random variable
Additivity of the Poisson distribution:
X ~ P(λ1)，Y ~ P(λ2)，则X+Y ~ P(λ1+λ2)Additivity of the normal distribution:
X ~ N(μ1，σ1^2)，Y ~ N(μ2，σ2^2)，且X、Y相互独立，则X+Y ~ N(μ1+μ2,σ1^2+σ2^2)minimax distribution
X1,X2...Xn相互独立，分布函数为：FXi(x)，Y = max{X1,X2...Xn},Z = min{X1,X2...Xn},则FY(y)=FXi(x)连乘，FZ(z) = 1-(1-FXi(x))的连乘