Suppose $\begin{bmatrix}X\\Y\end{bmatrix} \sim N \left( \begin{bmatrix}0\\0\end{bmatrix}; \begin{bmatrix}\sigma^2&\rho\sigma\\\rho\sigma&1\end{bmatrix} \right)$. How would I compute the conditional density, distribution and expectation of $X$ conditional on $Y>0$?
My attempt for the density function: \begin{align*} f(y|y>0)&=f(y)/P(Y>0)&\\ &=2f(y),\quad &y>0\\ f(x|y>0)&=f(x,y)/f(y|y>0)\\ &=f(x,y)/2f(y),\quad &y>0\\ \end{align*}
Distribution function: \begin{align*} F(x|y>0)&=P(X\leq x|Y>0)\\ &=P(X\leq x,Y>0)/P(Y>0)\\ &=\left(\int_0^\infty\int_{-\infty}^yf(u,v)dudv\right)\bigg/\int_0^ \infty f(y)dy \end{align*}
Expectation: \begin{align*} \mathrm{E}(X|Y>0)&=\int_{-\infty}^\infty xf(x|y>0)dx\\ &=\int_{-\infty}^\infty xf(x,y)/2f(y),\quad y>0 \end{align*}
Is this correct?