How can I find this probability $P(X
Finding probability P(X
12
$\begingroup$
probability
statistics
probability-theory
random-variables
-
0What if both of them are exponentially distributed random variables? – 2014-02-18
2 Answers
12
Assuming both variables are real-valued and $Y$ is absolutely continuous with density $f_Y$ and $X$ has cumulative distribution function $F_X$ then it is possible to do the following
$ \Pr \left[ X < Y \right] = \int \Pr \left[ X < y \right] f_Y \left( y \right) \mathrm{d} y = \int F_X \left( y \right) f_Y \left( y \right) \mathrm{d} y $
Otherwise, as @ThomasAndrews said in a comment, it is case-by-case.
7
I think we can control everything by the following general solution.
Consider $Z:=X-Y$. Then, by putting condition on the value of X, we get
$\begin{align} P(X
You may also put a condition on the value of $Y$ to get a similar result. So, the solution of this problem depends on what you want.