Moments of a Gaussian function

 

The moments of a line shape are

    The average áxñ

    The variance ááx2ñ - áxñ2ñ requires the above integral for the average and

To calculate the skew and higher moments we require integrals of the type

We can think of the odd-powered moments as having to do with the assymmetry of the function. A Gaussian centered about zero is a symmetric function and so all of the odd moments are zero. This is also logical since the average of Gaussian centered on zero 

will be zero.

The width of the Gaussian is not zero, however. The integral

can be solved using the identity

where I0(a) is

and

Therefore

and

.

The mean square displacement is also proportional to the width of the Gaussian distribution.