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Below is a problem I just did. My question for MSE is not how to solve it - but I provide it to illustrate what exactly I am asking.

Suppose X,Y are bivariate normal random variables with $E[X] = 40$, $\mathrm{Var}(X) = 76$, $E[Y] = 30$, $\mathrm{Var}(Y) = 32$, and $\mathrm{Var}(X | Y = 28.5) = 57.$

Calculate $\mathrm{Var}(Y | X = 25)$.

Although I know very little about bivariate random variables, I was able to solve this problem because I have a formula:

$$\mathrm{Var}(Y | X = x) = \sigma_{Y}^2(1 - \rho^2).$$

I am not certain, but based on convention I assume $\rho$ = $\rho_{X,Y}$ = $\frac{\mathrm{Cov}(X,Y)}{\sigma_X \sigma_Y}$.

Looking at the information given and my formula, I saw I could use the second formula to solve for $\rho$, and then re-use the formula to find the desired value. This is when I realized - the question in no way depends on the values of the conditioning variables ($Y = 28.5, X=25)$. This seemed strange to me. Keep in mind, my solution is just number crunching for me, I don't have a lot of background knowledge to provide intuition.

Can someone explain to me how this is intuitive that the function $f(x) = \mathrm{Var}(Y | X = x)$ is a constant function?

In my head when I picture a bivariate normal distribution I see what looks like an ant-hill centered over (0,0) in $\mathbb{R}^2$ (yes, technically I'm picturing a standard-bivariate normal). But then if I consider the cross sections cut out by fixing values of $X$, it seems the ones closer to the origin have a bigger hump - thus less variance? Is each cross section for different values of $X$ actually just like.. a scaling of the others? Thus variance stays fixed? Was this intentional in the construction of bivariate normals?

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2 Answers 2

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A way of seeing this is to consider how to generate pairs of random values from a bivariate Normal distribution with $X \sim N(\mu_X,\sigma^2_X)$ and $Y \sim N(\mu_Y,\sigma^2_Y)$ and covariance $\sigma_{X,Y}=\rho \sigma_{X}\sigma_{Y}$ between $X$ and $Y$.

One approach is to calculate the parts of $Y$ that depend on $X$ and do not depend on $X$ separately, and then add them together, which you can do as this is a bivariate Normal. The conditional variance of $Y$ given $X=x$ is then just the variance of the part of $Y$ that does not depend on $X$, and naturally this is not affected by the particular value $x$ that $X$ takes.

As a working algorithm:

  • Generate random values for $X\sim N(\mu_X,\sigma^2_X)$ using your favourite piece of software
  • Then $\rho \frac{\sigma_{Y}}{\sigma_{X}}X$ has mean $\rho \frac{\sigma_{Y}}{\sigma_{X}}\mu_X$ and variance $\rho^2 \sigma_{Y}^2$ and the covariance between $X$ and $\rho \frac{\sigma_{Y}}{\sigma_{X}}X$ is $\rho \sigma_{X}\sigma_{Y}$
  • Now generate random values for $Z\sim N\left(\mu_Y-\rho \frac{\sigma_{Y}}{\sigma_{X}}\mu_X,(1- \rho^2)\sigma_{Y}^2\right)$ independent of $X$ so the covariance between $X$ and $Z$ is $0$
  • Let $Y= Z+\rho \frac{\sigma_{Y}}{\sigma_{X}}X$. This has the result of $Y\sim N(\mu_X,\sigma^2_X)$ and $(X,Y)$ having a bivariate Normal distribution with covariance $\rho \sigma_{X}\sigma_{Y}$, which is what you are aiming for.

This means $\mathrm{Var}(Y \mid X = x) = \mathrm{Var}(Z)=(1- \rho^2)\sigma_{Y}^2$, which does not depend on the value of $x$

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You are correct that the conditional variance of $Y$ given $X=x$ does not depend on $x$. [Note however that the conditional mean does depend on $x$.]

Regarding your mental model: you are picturing the joint density, which takes into account the randomness in $X$. The reason why the cross sections for fixed $X$ seem "smaller" as $X$ is farther from its mean is partly due to the fact that it is less and less likely for $X$ to be farther and farther from the mean. However, the conditional distribution is not just the cross section of the joint density; note that you still have to renormalize according to the randomness in $X$. (For discrete variables, recall $P(Y=y \mid X=x) = P(Y=y, X=x) / P(X=x)$.) It turns out that after renormalizing to take account for the likelihood of seeing $X=x$, the conditional variances are the same value $\sigma_Y^2 (1-\rho)^2$ in the case of the bivariate normal distribution.

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