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In Probability and uniform distribution, the following equation is used:

$P(X_1>X_2) = \displaystyle \int_{0}^1 P(X_1>X_2 | X_2=x) f_{X_2}(x) dx$

How the equation is derived? Base on which definition or theorem?

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It follows from conditioning on $X_2$. In the case when $X_2$ is a discrete random variable, we have

\begin{align}P(X_1>X_2) &= \sum_{x_2} P(X_1 > X_2, X_2=x_2)\\ &= \sum_{x_2} P(X_1>X_2 \mid X_2=x_2) P(X_2=x_2) \\ &= \sum_{x_2} P(X_1 > x_2) P(X_2=x_2). \end{align}

When $X$ is a continuous random variable, the idea is the same, but we replace $P(X_2=x_2)$ with the density of $X_2$, and replace the sum with an integral. This gives the expression you wrote in your question. There are some technical issues with doing this, but I think you are at a point where you can overlook that.

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