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相关正态:P(X0,Y0)P(X\ge 0,Y\ge 0)

联合分布概率计算

专题
Probability / 概率
难度
L4

题目详情

Assume that XX and YY have joint normal distribution, that each of X,YN(0,1)X,Y\sim N(0,1) and that their correlation is 12\frac{1}{2} . Calculate P(X0,Y0)\mathbb{P}(X\geq 0,Y\geq 0) .

解析

对二维标准正态且相关系数为 ρ\rho,有经典公式

P(X0,Y0)=14+arcsin(ρ)2π.\mathbb{P}(X\ge 0,Y\ge 0)=\frac14+\frac{\arcsin(\rho)}{2\pi}.

ρ=1/2\rho=1/2arcsin(1/2)=π/6\arcsin(1/2)=\pi/6,所以

P(X0,Y0)=14+112=13.\boxed{\mathbb{P}(X\ge 0,Y\ge 0)=\frac14+\frac{1}{12}=\frac13}.