F Distribution: E(x) and Var(x) without Beta Distribution Transformation

Date: 07/07/2010 at 09:13:59
From: julie
Subject: finding E(x) and Var(x) when x has F diastribution

How do you find the expected value and the variance of the F distribution
using its probability density functions (PDFs)?

I integrated the messy xf(x) from 0 to infinity, but cannot do anything
after that.

Date: 07/07/2010 at 10:58:51
From: Doctor George
Subject: Re: finding E(x) and Var(x) when x has F diastribution

Hi Julie,

Thanks for writing to Doctor Math.

This article from the archive outlines how to do it.

  http://mathforum.org/library/drmath/view/70598.html 

Write again if you need more help.

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

Date: 07/07/2010 at 15:33:20
From: JULIE
Subject: finding E(x) and Var(x) when x has F diastribution

I looked at the archive. Can you please give me some hint about what the
new degrees of freedom should be?

Date: 07/07/2010 at 21:26:59
From: Doctor George
Subject: Re: finding E(x) and Var(x) when x has F diastribution

Hi Julie,

If your F distribution has m and n degrees of freedom, then it includes a
term that looks like this:

    x^(m/2 - 1)

When you form the integral to compute the mean, you get another x from the
xf(x). Combining them gives you this:

    x^(m/2)

We want to get back to the previous form, so we define m' so that

    m'/2 - 1 = m/2
      m' - 2 = m

So substitute m = m' - 2 into your integral. Now use properties of
exponents and the gamma function to obtain a form that starts looking like
the PDF in terms of m'. Then define n' by focusing on the form of the
gamma terms. Finally, you will get a constant times the PDF in terms of m'
and n'. That constant gets moved outside the integral, and the result then
just evaluates to 1.

There is quite a bit to this. See how far you can get now.

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

Date: 07/07/2010 at 21:58:12
From: julie
Subject: finding E(x) and Var(x) when x has F diastribution

I understand what the new m will be. I still cannot figure out what the
new n will be.

Please help.

Date: 07/08/2010 at 11:28:00
From: Doctor George
Subject: Re: finding E(x) and Var(x) when x has F diastribution

Hi Julie,

As I wrote in the archive article, this is not a simple problem, and the
complete details are hard to follow in a text format. Let's start over and
try to come up with something easier to follow.

The F distribution is related to the beta distribution, which is much
easier to work with. So let's do a change of variables in your integral.

    x = (n/m) * w / (1 - w)

This may look a little imposing if you have not seen such a change of
variables before, but it actually simplifies quite a bit.

The new integrand will look much like the integrand for the mean of the
beta distribution. From there, it should be much simpler to see how to
select values for m' and n' so that the integrand becomes a constant times
the PDF of the beta distribution.

See if that works any better for you.

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

Date: 07/09/2010 at 07:14:05
From: Doctor George
Subject: Re: finding E(x) and Var(x) when x has F diastribution

Hi Julie,

It has been bugging me that it is so hard to do this problem without using
the beta distribution transformation. I finally came up with a better
method working directly with the F distribution.

Perform this change of variables in the integral for the mean:

        n  m'
    x = -  -- x'
        m  n'

Many things will simplify, and you should start to see the form of the PDF
for x' begin to emerge. When that happens, the needed relationships
between m and n, and m' and n', will become apparent.

    m'   m                  n'   n
    -- = - + 1              -- = - - 1
    2    2                  2    2

This leads to

    m = m' - 2
    n = n' + 2

After you make those substitutions and use a property of the gamma
function, the final result should appear. You will get a constant (which
is the mean) times the integral of the PDF in m', n', and x' (which is
just 1).

Then use exactly the same technique when finding E(X^2) to obtain the
variance.

I hope this is more helpful.

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