Calculate $\mathbb E(Y^2\mid X)$

Question

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 $\mathbb E(Y^2\mid X)$.

Answer 1

The important fact to use is that $X$ and $Y$ are bivariate normal. Multivariate normal distributions have nice conditional distributions.

In particular, if you instead let $X$ and $Z$ be i.i.d. standard normal, then you can check that if you re-define $Y$ as $$\frac{1}{\sqrt{5}} Y := \rho X + \sqrt{1 - \rho^2} \cdot Z$$ with $\rho = -\frac{2}{\sqrt{5}}$, then $X$ and $Y$ are bivariate normal with the expectations/variances/covariance given in the question. (Check this.)

With this formulation, the conditional distribution of $Y$ given $X$ is easy to obtain. Given $X=x$, we have $$(Y \mid X=x) \overset{d}{=} \sqrt{5} \rho x + \sqrt{1 - \rho^2} \cdot Z \sim N(\sqrt{5} \rho x, 1-\rho^2)$$ so $\text{Var}(Y \mid X) = 1-\rho^2$ and $E[Y \mid X] = \sqrt{5} \rho X$, from which you can compute $E[Y^2 \mid X]$.

Answer 2

Recall that $Var(Y|X) = \mathbb{E}[Y^2|X] - \mathbb{E}^2[Y|X].$ We have expressions for the LHS and the second term of the RHS, see here: https://en.wikipedia.org/wiki/Multivariate_normal_distribution (under "Conditional distributions"). In particular, conditional on $X,$ $Y\sim N(-2X, 1).$

This should allow you to find $\mathbb{E}[Y^2|X].$

Answer 3

We can find some $c$ such that the Gaussian random variables $Y-cX$ and $X$ are uncorrelated hence independent. Then write $$ \mathbb E\left(Y^2\mid X\right)=\mathbb E\left(\left(Y-cX+cX\right)^2\mid X\right), $$ expand the square: we get a sum of three terms which consist of products of terms which are either $\sigma(X)$-measurable or independent of $X$.

Answer 4

$Y|X\sim N(\mu_Y+\rho\frac{\sigma_Y}{\sigma_X}(x-\mu_X);\sigma_Y^2(1-\rho^2))$

And

$\mathbb{E}[Y^2|X]=\mathbb{V}[Y|X]+\mathbb{E}^2[Y|X]$