Unique properties of Gaussians

  • Thread starter mnb96
  • Start date
710
5
Hello,
let's suppose we are given a Gaussian function [tex]f(x)=Ae^{-ax^2}[/tex] (where a,A are real scalars and a is positive)

Is it possible to (dis)prove that the following identity is true only for Gaussians?

[tex]\frac{\partial^2}{\partial x^2} log f(x) = const[/tex]

Thanks!
 
Last edited:
319
0
If you solve the differential equation you end up with
[tex] f(x) = A e^{const\. x^2 + B x}.[/tex]
So provided the constant is negative, you get a gaussian with a shift in the variable.
 

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