Moments
The raw moments are then given by:
where is the gamma function. Thus the first few raw moments are:
where the rightmost expressions are derived using the recurrence relationship for the gamma function:
From these expressions we may derive the following relationships:
Mean: which is close to for large k.
Variance: which approaches as k increases.
Skewness:
Kurtosis excess:
Large n approximation
We find the large n=k+1 approximation of the mean and variance of chi distribution. This has application e.g. in finding the distribution of standard deviation of a sample of normally distributed population, where n is the sample size.
The mean is then:
We use the Legendre duplication formula to write:
- ,
so that:
Using Stirling's approximation for Gamma function, we get the following expression for the mean:
-
And thus the variance is: