# Find distribution with given mean and variance

I have an exercise with which I'm currently struggling:

Consider a random sample $\big\{X_n\big\}_{n=1}^N$ from a normal distribution having mean $\mu$ and variance $\sigma^2$. What is the distribution of $\frac{\sum_{n=1}^N (X_n - \overline{X}_N)^2}{\sigma^2}$?

Could anyone give me some hints in the right direction on how to achieve this?

• Do you know anything about the distribution of the sample variance? Commented Nov 8, 2016 at 7:40
• Not quite. However, as far as i've seen during my reading on wikipedia, the variance of $\overline{X}_N$ should be $\frac{s}{\sqrt{N}}$ with $s$ the sample standard deviation... Commented Nov 8, 2016 at 7:50
• Are you studying from a text book or from lecture notes? If so, there's probably something there about the distribution of (a multiple of) $S^2$. Commented Nov 8, 2016 at 7:55
• Hmm, just required work ahead of class.So there are basically no lecture notes nor a book... Commented Nov 8, 2016 at 7:57
• "Required work" sounds like you should have enough information to answer the question. Otherwise, have a look at en.wikipedia.org/wiki/… Commented Nov 8, 2016 at 8:05