Forumla for finding conditional variance

Question

I need to find the conditional variance $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, given that $X_1\sim N(0,1)$ and $X_2+X_3\sim N(0,2+2\gamma)$. The covariance between X1, X2+X3 is $\rho$.

From this link: http://athenasc.com/Bivariate-Normal.pdf,under the section of properties of jointly normal variables, the conditional variance $\mathop{\mathrm{Var}}(X|Y)$ is equal to $(1-\rho^2)\sigma_x^2$.

If you apply the same thing to the conditional variance of $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, you should get $(1-\rho^2)*1$, but when I find the conditional probability density function of $X_1|X_2+X_3$, the value I get from the density function is ($2+2\gamma-\rho^2)/2+2\gamma$.

Not sure why there is a such a difference. Need some guidance.

Answer 1

If $(X,Y)$ is centered normal, $\mathrm{var}(X\mid Y)=(1-r^2)\sigma_X^2$, where $r$ is the correlation of $X$ and $Y$ (not the covariance). In your case, $X=X_1$, $Y=X_2+X_3$, $\sigma_X^2=1$ and $\sigma_Y^2=2+2\gamma$, hence, if the covariance of $X$ and $Y$ is $\rho$, the correlation is $r=\rho/\sqrt{2+2\gamma}$ and the conditional variance is $(1-r^2)\sigma_X^2$, that is, $1-\rho^2/(2+2\gamma)=(2+2\gamma-\rho^2)/(2+2\gamma)$.

Although this answer follows your notations, it should be noted that to call $\rho$ a covariance is definitely a bad idea, since the letter is universally used for correlations.