Marginal PDF for Sum of Squares: Identifying Distribution and Calculation Method

[Fermat1]

Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks

 

[I like Serena]

Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks

The chi-square distribution is defined as $$\chi_n^2 = \sum_{i=1}^n Y_i^2$$ where $Y_i \sim N(0,1)$.
We need to rewrite the expression to standard normal distributions to relate it to the chi-square distribution.
The first step has already been done, since $X_i-\overline X \sim N(0,\sigma^2)$.