设X~N(0,σ^2),N(0,4σ^2),而且P{X≤1,Y≤-2}=1/4,则概率P{X>1,Y>-2}=?

Question

设X~N(0,σ^2),N(0,4σ^2),而且P{X≤1,Y≤-2}=1/4,则概率P{X>1,Y>-2}=?

Answer 1

P(X≤1 and Y≤-2)=1/4
P(X≤1)P(Y≤-2)=1/4
P(Z≤1/σ)P(Z≤-1/σ)=1/4
[1-Φ(-1/σ)]Φ(-1/σ) = 1/4
[Φ(-1/σ)]^2-Φ(-1/σ) +1/4 =0
Φ(-1/σ) = 1/2
P(X>1,Y>-2)
=P(X>1)P(Y>-2)
=P(Z>1/σ)P(Y>-1/σ)
=Φ(-1/σ) [ 1-Φ(-1/σ)]
= 1/4