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Suppose $X\sim\mathrm{Exp} $ and $Y$ is a non negative independent random variable. Does $P(X-Y>t\mid X>Y)=P(X>t)$, where $t$ is a non negative real number. I think it is true as it seems to be can treat the random variable $Y$ as some constants but not sure whether it is correct.

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The claim is correct. This is a generalization of the memoryless property of exponentials.

Observe: $$P(X-Y > t|X>Y) = \int_{0}^{\infty}P(X>y+t|X>y,Y=y)dF_Y(y)$$ $$ = \int_{0}^{\infty}P(X>y+t|X>y)dF_Y(y)$$ due to independence $$ = \int_{0}^{\infty}P(X>t)dF_Y(y)$$ due to memoryless property of exponential $$ = P(X>t)\int_{0}^{\infty}dF_Y(y) = P(X>t)$$

QED

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  • $\begingroup$ your first integral should integrate with respect to $y$ instead of $F_Y(y)$, aren't it? $\endgroup$ – Mathematics Dec 21 '12 at 5:24
  • $\begingroup$ If Y has a pdf say $f_y(y)$ then replace $dF_y(y)$ with $f_y(y)dy$. If not, then we have to use the Riemann Stieltjes form which is what I have written. $\endgroup$ – Gautam Shenoy Dec 21 '12 at 12:47

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