# Express moment / expectation value in lower order expectation values

1. Jan 6, 2012

### 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.

2. Jan 6, 2012

### mathman

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

3. Jan 6, 2012

### flux2

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

Cheers

4. Jan 6, 2012

### 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>