# Conditional independence of differences between normal random variables

$X_1, X_2, X_3, X_4$ are independent, normally distributed random variables with different means and variances. Let $$Y_1 = X_1 - X_2 \\ Y_2 = X_2 - X_3 \\ Y_3 = X_3 - X_4 \\$$ Is it true that $$P(Y_1 > 0 \mid Y_2 > 0, Y_3 > 0) \stackrel{?}{=} P(Y_1 > 0\mid Y_2>0)$$ Note that $Y_1 \perp Y_3$.

I think I know the answer to this, but just want some other opinions.

$Y_1$ is independent of $Y_3$, but not conditionally independent given $Y_2$, so I wouldn't expect this to be true. Try an example where $X_1, \ldots, X_4$ have the same distribution. Note that e.g. $(Y_1 > 0) \text{ and } (Y_2 > 0) \text{ and } (Y_3 > 0)$ iff $X_1 > X_2 > X_3 > X_4$, which has probability $1/4! = 1/24$ since each of the $4!$ possible orderings is equally likely. Similarly for other events. I get $$P(Y_1 > 0 | Y_2 > 0, Y_3 > 0) = \dfrac{P(Y_1 > 0, Y_2 > 0, Y_3 > 0)}{P(Y_2 > 0, Y_3 > 0)} = \dfrac{1/24}{1/6} = \dfrac{1}{4}$$ $$P(Y_1 > 0 | Y_2 > 0) = \dfrac{1/6}{1/2} = \dfrac{1}{3}$$