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/
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