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Josh

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- Thread starter joshthekid
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Josh

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Because your random variables are independent, ##p_{X,Y}(x,y) = p_X(x)p_Y(y)##, where ##p_X## and ##p_Y## are the pdfs of your individual random variables.

Then you just integrate over the region A that is the set of points where X < Y, and you have P(X < Y).

I'll do the exponential distribution as an example. ##p_X(x) = \alpha e^{-\alpha x}##, and ##p_Y(y) = \beta e^{-\beta y}##. So the joint pdf is ##p_{X,Y}(x,y) = \alpha \beta e^{-\alpha x - \beta y}##. Now you integrate:

##P(X < Y) = \int_0^\infty \int_0^y \alpha \beta e^{-\alpha x - \beta y}\,dx\,dy = \alpha \beta \int_0^\infty e^{-\beta y} {1 - e^{-\alpha y}\over \alpha}\,dy = 1 - {\beta \over \alpha + \beta} = {\alpha \over \alpha + \beta}.##

So that's the answer.

- #3

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thanks Eigenperson!

Josh

Josh

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