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Say $X, Y$ are standard normals with correlation $\rho$.

How do I calculate conditional probabilities such as $P(X \le x \mid Y \le y)$?

I don't have any assumptions on their joint distribution, so what are my options?

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    $\begingroup$ You can't find the desired probability because it depends on the unknown joint distribution which you cannot assume to be jointly normal as you say. A huge collection describing how two normal random variables can fail to be jointly normal can be found in this answer by Moderator cardinal $\endgroup$ – Dilip Sarwate Dec 22 '18 at 18:55
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    $\begingroup$ Possible duplicate of Is it possible to derive joint probabilities from marginals with assumptions about the conditionals? $\endgroup$ – Xi'an Dec 22 '18 at 20:38
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    $\begingroup$ If you know the correlation, then you know more than just the marginals. However, you've stopped short of fully specifying the joint distribution: the marginals could both be $\operatorname N(0,1)$ and the correlation $\rho$ and the joint distribution bivariate normal, but the marginals could both be $\operatorname N(0,1)$ and the correlation $\rho$ even if the joint distribution differs in any of various ways from the bivariate normal. $\endgroup$ – Michael Hardy Dec 23 '18 at 0:14
  • $\begingroup$ You should look into copulas. Try to do the calculations with different copulas all having the same correlation $\rho$ $\endgroup$ – kjetil b halvorsen Dec 23 '18 at 0:42
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    $\begingroup$ This question might be slightly different then the one referenced as duplicate. That question is very general. This question is more specific by mentioning additional conditions. Although from that other question you might deduce that there are an infinite number of ways to get two marginal normal distributions and a specific correlation. $\endgroup$ – Sextus Empiricus Dec 23 '18 at 7:59

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