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∗pdf

_{A}(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)=pdf

_{A}(A)

hence by solving this for pdf

_{A}(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