Conditional marginal distribution of conditional bivariate normal distribution

Question

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 $Y$?

Answer 1

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 $(X,Y)$.