How to get the distribution of $\frac{1}{n} \sum_i x_i^2$ $X$ follows a normal distribution $X \text{~} N(\mu, \sigma^2) $.
And there are $n$ samples. Then what is the distribution of $$\frac{1}{n}  \sum_i x_i^2$$
I do understand $\frac{\sum_i x_i}{n} \text{~} N(\mu, \frac{\sigma^2}{n})$, but don't know how to apply them here. Anybody who can help please?
 A: If $X_1,...,X_n \sim \text{IID N}(\mu, \sigma^2)$ then this statistic has a non-central chi-squared distribution.  To see this, we observe that $X_1/\sigma,...,X_n/\sigma \sim \text{IID N}(\mu/\sigma, 1)$ so that:
$$\frac{1}{n} \sum_{i=1}^n X_i^2 = \frac{\sigma^2}{n} \sum_{i=1}^n \Big( \frac{X_i}{\sigma} \Big)^2 \sim \frac{\sigma^2}{n} \cdot \text{N-C Chi-squared} \Big( DF = n, NCP = \frac{n \mu^2}{\sigma^2} \Big).$$
A: It is (nearly) a noncentral chi-squared distribution by definition.
The difference is in a scaling factor.

Definitions
Both the central and noncentral chi-squared distribution with $n$ degrees of freedom are the distribution of the sum of the squares of $n$ independent distributed normal distributed variables. 
$$\sum_{i=1}^{n} X_i^2$$


*

*For the (central) chi-squared distribution you have $$X_i \sim N(0,1)$$. 

*For the noncentral chi-squared distribution you have $$X_i \sim N(\mu_i,1)$$ and $\lambda = \sum_{i=1}^{n} \mu_i^2$ is the non-centrality parameter. 


Transformation
For more general variables $X_i \sim N(\mu_i,\sigma_i^2)$ you have that
$$\sum_{i=1}^n \frac{X_i^2}{\sigma_i^2} \sim \chi^2 \left( n,\sum_i (\mu_i / \sigma_i)^2 \right)$$
is a noncentral chi-squared distributed variable. So your case is a transformed version of it:
$$\frac{1}{n} \sum_{i=1}^n X_i^2 = \frac{\sigma^2}{n} \sum_{i=1}^n \frac{X_i^2}{\sigma_i^2}$$

The actual distribution function
Getting the actual expression for the distribution of the noncentral chi-squared distribution is not as easy as for the central chi-squared distribution (which can be derived easily using symmetry arguments). It will eventually be represented in terms of analytical functions (e.g. the modified Bessel function of the first kind).
