Calculate $\mathbb E(Y^2\mid X)$ 
Let $X,Y$ be random variables with the total normal distribution such that $$\mathbb EX=\mathbb EY=0, \operatorname{Var} X=1, \operatorname{Var} Y=5, \operatorname{Cov}(X,Y)=-2$$Calculate  $\mathbb E(Y^2\mid X)$

From this task I can also calculate


*

*$\mathbb E(XY)=\operatorname{Cov}(X,Y)+\mathbb EX \cdot \mathbb EY=-2$

*$\mathbb EX^2 =\operatorname{Var}X+(\mathbb EX)^2 =1$

*$\mathbb EY^2=5$
However, I know that $$\mathbb E(Y^2\mid X)=\int_{\Omega} Y^2 d \mathbb P_X$$ so this information is unhelpful and I don't know how to calculate $\mathbb E(Y^2\mid X)$.

$Y|X\sim N(\mu_Y+\rho\frac{\sigma_Y}{\sigma_X}(x-\mu_X);\sigma_Y^2(1-\rho^2))$
And 
$\mathbb{E}[Y^2|X]=\mathbb{V}[Y|X]+\mathbb{E}^2[Y|X]$
