# Conditional expectation of an exponential variable

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 $$E[X | X > Y]$$ = $$\int_0^\infty X f_{X|X>Y}(x)dx$$ = $$\int_0^\infty X (1-f_{X|X = $$\int_0^\infty Xdx$$ - $$\int_0^\infty X f_{X|X = $$1 - \frac{1}{\lambda_x +\lambda_y}$$.

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.

• 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 $1$)? – whuber Feb 7 at 16:02