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Suppose that $A$ and $B$ are independent random variables, $P$ and $F$ are the probability and the cumulative distribution functions. Can we write this ?

$$P(A>B) = P(A-B>0)$$ $$ = 1 - F_{A-B}(0)$$

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  • $\begingroup$ Your title doesn't match what you ask in the body of your post. Please edit to make them consistent. $\endgroup$ – Glen_b Jul 25 '17 at 11:17
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If you define $F_{A-B}(x)=P(A-B \leq x)$, which is indeed the cumulative distribution function of the variable $A-B$, then the answer is straightforward, isn't it?

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