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>

 

[CellCoree]

I am happy to provide a response to this content. The equation you are looking for is known as the "moment/expectation value relationship" and is a fundamental concept in statistical mechanics. It states that the expectation value of a higher order moment (in this case, <x6>) can be expressed in terms of lower order moments (specifically, <x>6 and <x>4). This relationship is useful in simplifying complex calculations and understanding the behavior of a system.

To understand this relationship, let's first define what moments and expectation values are. Moments are statistical quantities that describe the shape and distribution of a set of data. In this case, <x6> represents the 6th moment of the data set, which is a measure of the spread of the data around its average value. On the other hand, expectation values are the average value of a quantity in a given system. In this case, <x>6 and <x>4 represent the average values of the 6th and 4th powers of the variable x, respectively.

Now, the proof of the moment/expectation value relationship is based on the definition of the standard deviation, which is given by the square root of the 2nd moment minus the square of the 1st moment. Using this definition and some mathematical manipulation, we can show that <x6> is equal to <x>6 plus 15 times the standard deviation squared times <x>4. This is exactly the equation you were looking for.

In summary, the moment/expectation value relationship allows us to express higher order moments in terms of lower order moments, making it a powerful tool in statistical mechanics. I hope this explanation helps you understand the equation and its significance. Keep exploring and asking questions, and best of luck in your research!