Express moment / expectation value in lower order expectation values

[flux2]

Hello everybody,

I'm looking for a proof of the following equation:

<x⁶> = <x>⁶+15s²<x>⁴

where the brackets denote an expectationvalue and s is the standard deviation.

Thanks in advance!

 

[mathman]

It is not true in general. For example if <x> = 0, <x⁶> will not = 0 unless x = 0 itself.

 

[flux2]

It is not true in general. For example if <x> = 0, <x⁶> will not = 0 unless x = 0 itself.

Sorry, I forgot to mention. It's a first order Taylor approximation.

Thanks for the reply,

Cheers

 

[flux2]

We have for example:

<x²> = <x>²+s²


<x⁴> - <x²>² ≈ 4s²<x>² (to first order)

Now combining both equations yields:

<x⁴> = <x>⁴+6s²<x>²

Unfortunately this doesn't work that easily for <x⁶>