Independent Probablilty Distributions

Date: 12/15/2005 at 07:35:57
From: Phil
Subject: Independent Probablilty Distributions

I have two independent random variables A and B with known continuous
density and distribution functions and their pdf's overlap.  How do I
calculate the probability that variable A will be lower than variable B?

I've looked at the intersections between their pdf and cdf and mused 
about the ratio between the area under the cdf but can't think how to 
prove this.

Many thanks,

Phil

Date: 12/16/2005 at 15:51:49
From: Doctor George
Subject: Re: Independent Probablilty Distributions

Hi Phil,

Thanks for writing to Doctor Math.

Since A and B are independent, their joint density is

  f(a)f(b)
   A   B 

To find P(A<B) we need to integrate the joint density over the half
plane a<b.  Here is one way to do this.

             oo   b
            /    /
  P(A<B) =  |    |  f(a)f(b) da db
            /    /   A   B
            -oo  -oo

             oo
            /
         =  |  F(b)f(b) db
            /   A   B
            -oo

If the joint density is non-zero over only some region of the ab plane
you can set the limits of integration accordingly.  As you look for 
the easiest way to carry out the integration it can be useful to note 
that

  P(A<B) = 1 - P(B<A)

(Note that P(A=B) = 0 since A and B are continuous.)

Finally, it can be helpful to think of the solution like this

  P(A<B) = E[F(B)]
              A  

where E denotes expected value.  Viewed in this way, P(A<b) for some
particular value b is averaged over all possible values of b.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/