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Let $(X,Y]$ be a non degenerate random interval and the probability is that:
$P(X\leqslant Y)=1$ , $P(X=Y)=0$.
We define $E[Y-X]$ as its expected length.
Show that $E[Y-X]=\int P(X < t\leqslant Y)dt$

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    And the identity between random variables $Y-X=\int\limits_{\mathbb R}\mathbf 1_{X\lt t\leqslant Y}\mathrm dt$.2012-05-01

1 Answers 1

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(Answered so that the question may be closed.)

Hint: Use Tonelli's theorem and the identity between random variables $Y−X=\int\limits_\mathbb R\mathbf 1_{X\lt t\leqslant Y}\mathrm dt$.