How can I find this probability $P(X
Finding probability $P(X
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$\begingroup$
probability
statistics
probability-theory
random-variables
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4In general, just knowing they are independent isn't useful, since $X$ might be a random number from $-1$ to $0$ and $Y$ might be a random number from $1$ to $2$, or visa versa, giving you probabilities anywhere between $1$ and $0$. If they are identical and independent continuous random variables, then the probability will be $1/2$. (Continuity implies $P(X=Y)=0$ and identical implies $P(X
Y)$ .) – 2012-12-17 -
0What if both of them are exponentially distributed random variables? – 2014-02-18
2 Answers
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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.
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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.