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Given two normally distributed random variables $ X_1 \sim N(\mu_1, \sigma_1^2) $ and $ X_2 \sim N(\mu_2, \sigma_2^2) $, with known correlation $ \rho_{12} $ how can I find:

$$ \begin{align} & \ \ \ \ \ \ P(X_1 > x > X_2) \\ & \equiv P((X_1 > x)\ \cap (X_2 < x)) \\ & \equiv P((X_1 > x)\ \cap (X_1 - X_2 > 0)) \end{align} $$

I know $ (X_1 - X2) \sim N(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2 + 2\rho_{12}\sigma_1 \sigma_2) $, but I don't know to calculate the joint probability. Do I need to derive the correlation of $X_1$ and $(X_1 - X_2)$?

Thanks!

Edit:

$X_1$ and $X_2$ are themselves a known linear combination of independent normal distributions, hence why their correlation is known. So I can also calculate the distribution of the components they have in common, and the distributions of their components that are unique. Perhaps this helps.

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  • $\begingroup$ The question can not be answered without assuming that the joint distribution of $(X_1,X_2)$ is normal or any other assumptions on the joint distribution. If the joint distribution is normal then the usual way to solve such problems is to find a linear transform that makes the two components of the random vector independent (i.e., uncorrelated in this case) and identically distributed. $\endgroup$ – Viktor Oct 14 '17 at 15:50
  • $\begingroup$ Okay, let's assume the joint distribution is normal. Can you expand on your last sentence? What are the two components of the random vector you are referring to? $\endgroup$ – Albeit Oct 14 '17 at 16:02

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