$X,Y$ are standard normal random variables, the correlation is $0.5$. Given $X$, what's the conditional distribution of $Y$?

Question

The following is an interview question:

$X,Y$ are standard normal random variables, the correlation is $0.5$. Given $X$, what's the conditional distribution of $Y$?

I have no idea how to start this question at all.

Any hint is appreciated.

Answer 1

I agree with Aaron that without being familiar with some necessary background material, the answer isn't going to make much sense. However, it is still worth mentioning that there is no consistent answer to this question.

This is a usual caveat / nuance when dealing with several normal random variables: you must know in advance that they are jointly Gaussian, or else have it be some indirect consequence of what you do know. Otherwise, strange things can happen.

In this case, if $X$ and $Y$ happen to be jointly Gaussian, then the conditional distribution is of course Gaussian (this is a theorem that one ought to be able to at least understand its meaning, see Aaron's comment to user658409's answer) and its parameters can indeed be deduced as hinted by user658409's answer.

On the other hand, if $X$ is a standard Gaussian and $Z$ is a discrete random variable that is independent of $X$ and satisfies $$ Z = \begin{cases} 1, & \text{with probability } 0.75,\\ -1, & \text{with probability } 0.25, \end{cases} $$ then one can readily check that $Y=ZX$ is also a standard Gaussian, the correlation between $X$ and $Y$ is $0.5$, but the conditional distribution of $Y$ given $X$ is just discrete: $X$ with probability $0.75$, and $-X$ with probability $0.25$. This is a variation of a simple classical example.

Answer 2

You can just write out the distribution. It is a multivariate normal distribution with mean vector $\mu =\langle 0,0\rangle$ and covariance matrix $\Sigma=\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix}$. You can then just use the undergraduate definition of conditional distribution.