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I have trouble with conditional probabilities, therefore I am wondering if the following derivation is correct:

Both X and Y are exponential random variables, with $\lambda_x$ and $\lambda_y$ respectively as their rates, and are independent of each other. Then $E[X | X > Y]$ = $ \int_0^\infty X f_{X|X>Y}(x)dx$ = $ \int_0^\infty X (1-f_{X|X<Y}(x))dx$ = $ \int_0^\infty Xdx$ - $ \int_0^\infty X f_{X|X<Y}(x)dx$ = $ 1 - \frac{1}{\lambda_x +\lambda_y} $.

I remember the property $E[X | X < Y]$ = $E[min(X,Y)]$ = $\frac{1}{\lambda_x +\lambda_y} $. But it just seems wrong to me, that you get 1 - $E[X | X < Y]$ as answer for $E[X | X > Y]$, as if it is a probability. So I assuming that I am making an error.

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  • $\begingroup$ How did you arrive at "$\int_0^\infty X dx = 1$"? If you really mean capital $X,$ then this expression makes no sense; and if you mean lower case $x,$ then it diverges. Before you worry about this too much, could you explain why you believe $f_{X\mid X\gt Y} = 1- f_{X\mid X\lt Y}$? How would you interpret this for values of $f$ that exceed $1$ (which will happen when rates exceed $1$)? $\endgroup$ – whuber Feb 7 at 16:02

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