# Need Some Mathematical Guidance Regarding Random Variables

This is not a homework question but I project I am working on and need someone with more mathematical prowess than myself. I am using a computer program to draw random numbers from two independent distributions, x1 and x2, for two different cases and I want to establish a theoretical mathematical relationship for the probability that x1 will be less then x2. The first case is two different exponential distributions, i.e. exp(-αx), and the second from a power law, x^(-β), over a limited range a to b. I have been working on this for about a week so any help or guidance would be much appreciated. Thanks

Josh

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What you need to do is write down the joint probability distribution function. This is a function ##p_{X,Y}## in variables x and y such that for any region A, ##\iint_A p_{X,Y} = P((X, Y) \in A)##.

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}.##