# Paradox. Extracting probability density function from mean.

• Mar 14th 2014, 05:28 PM
Rafael
Paradox. Extracting probability density function from mean.
Is it possible to obtain probability density function of continuous distribution from it's mean?

1. No. Probability density function contains more information about distribution than it's mean. Particularly one can obtain same mean for distributions with different probability density functions.

2. Yes.
<A>=∫−∞∞A∗pdfA(A)dA

can be viewed as special case of Fredholm integral equation of the first kind
(Fredholm Integral Equation of the First Kind -- from Wolfram MathWorld)
where

f(x)=<A>
K(x−t)=A
ϕ(t)=pdfA(A)

hence by solving this for pdfA(A) we will obtain probability density function of the distribution from it's mean.

I can not find where am I doing mistake in my reasoning. It should be in the second point I think.
I will be thankful if you will provide solution for this paradox.

Regards,
Rafayel
• Mar 14th 2014, 05:41 PM
HallsofIvy
Re: Paradox. Extracting probability density function from mean.
It should be obvious that you can't get the probability density function from its mean because many different probability densities have the same mean.

The error in your (2) is that solutions to such a Fredholm equation are NOT unique.
• Mar 14th 2014, 05:44 PM
romsek
Re: Paradox. Extracting probability density function from mean.
Quote:

Originally Posted by Rafael
Is it possible to obtain probability density function of continuous distribution from it's mean?

1. No. Probability density function contains more information about distribution than it's mean. Particularly one can obtain same mean for distributions with different probability density functions.

2. Yes.
<A>=∫−∞∞A∗pdfA(A)dA

can be viewed as special case of Fredholm integral equation of the first kind
(Fredholm Integral Equation of the First Kind -- from Wolfram MathWorld)
where

f(x)=<A>
K(x−t)=A
ϕ(t)=pdfA(A)

hence by solving this for pdfA(A) we will obtain probability density function of the distribution from it's mean.

I can not find where am I doing mistake in my reasoning. It should be in the second point I think.
I will be thankful if you will provide solution for this paradox.

Regards,
Rafayel

Is it possible to find "a" probability density function from it's mean. Sure. Pick a density function of your choice and select parameters so that it's mean equals the number you are given.

Is it possible to find "the" probability density function from it's mean. No, because it isn't unique. For any given number there are infinitely many density functions having this mean.