## Find Probability distribution that matches given order statistic property

Given $0 < p < 1$, do there exist probability distributions for random variables $X,Y$ such that all three of following are true:

$$P(X < Y) = p$$ $$pdf (X) = f(x, p)$$ $$pdf (Y) = f(y, 1-p)$$

Of course the trouble is trying to keep $f$ same.

Or equivalently, find an $f$ that satisfies the following equation for all $0 < p < 1$ $$\int_{x=0}^{\infty} f(x,p) \int_{y=x}^{\infty} f(y, 1-p) dy~dx = p$$

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