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Let us assume $X,Y,Z$ three independent $U(0,1)$ random variables. Is it true that

$$P(Z < Y < X) = \int_{z = 0}^{1} \int_{y = z}^1 \int_{x=y}^1 dx dy dz $$

I know this is the approach for $P(Y < X)$, and I am wondering how to generalise this for an event with $n$ variables, e.g. $P(X_n < X_{n-1} < ... < X_1)$.

mth_mad
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Yes, this is the correct way to do it. And you can check your expression by doing the integral, which comes hout to have probability $1/n!$ for the $n$-variables case.

Mark Fischler
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  • "this is the correct way to do it" One can also "do it" with no integral at all, using a symmetry argument which is well worth knowing. – Did Dec 13 '16 at 23:52
  • Could you explain a little more about the symmetry argument? – mth_mad Dec 13 '16 at 23:55