Determining the distribution of Y based on $x̄$ and $S^2$

Question

Let $X1,...,Xn$ be a random sample from $N(μ,σ^2)$ and let $x̄$ and $S^2$ be the usual sample mean and sample variance. Define the random variable $$Y=c(x̄-μ)^2/S^2$$

Find c such that Y is a "named" distribution.

Question: I'm having trouble getting started. I know the formulas for $x̄$ and $S^2$ but I'm not sure if I should be using the density for a normal in any way. If someone could face me in the right direction that would be great.

Answer 1

Hint: That normal distribution $N(\mu,\sigma^2$) has its use. Remember that from Fisher's theorem we now that $\frac{\overline{X}-\mu}{\sigma}\sqrt{n}\sim N(0,1)$ and $(n-1)\frac{S^2}{\sigma^2}\sim\chi_{n-1}^2$. Also remeber that by definition, the random variable

$Y=\sum_{i=1}^nX_i^2$ where $X_i\sim N(0,1)$, $ \forall i=1,...,n$,

has a distribution $\chi_n^2$. Now try to arrange those facts to get that "named" distribution.