# Conditional marginal distribution of conditional bivariate normal distribution

I have a bivariate normal distribution$$(X, Y)\sim N(\mu_{x}, \mu_{y}, \sigma_{x}^2, \sigma_{y}^2, \rho)$$ My question is : when $$X > k$$ ($$k$$ is a constant),how to get the distribution of $$Y$$? Can anyone tell me how to solve it? For exaple, let $$(X, Y) \sim N(0, 0, 1, 1, 0.7)$$ when $$X > 1$$, the distribution of $$Y$$?

Using usual notation, the conditional (truncated) distribution $$Y\mid X>k$$ for some fixed $$k$$ is given by

\begin{align} f_{Y\mid X>k}(y)&=\int_k^\infty\frac{f_{X,Y}(x,y)}{P(X>k)}\,dx \\\\&=\frac{1}{P(X>k)}\int_k^\infty f_{Y\mid X=x}(y\mid x)f_X(x)\,dx\qquad,\,y\in\mathbb R \end{align}

You can now find this density explicitly given any joint distribution $$(X,Y)$$.

• I'm sorry I can't understand well about the expression you showed. Can you recomment relevant materials？ Please – riskingitall Nov 21 '18 at 11:47
• @riskingitall $f_{X,Y},f_X,f_{Y\mid X}$ are respectively the joint density of $(X,Y)$, the marginal density of $X$ and the conditional density of $Y\mid X$. Which expression is unclear? – StubbornAtom Nov 21 '18 at 12:13
• wow, I got it! You're a genius.Thank you very very much.This question has been bothering me for a long time. – riskingitall Nov 21 '18 at 12:30