Let ( X 1 , , X n ) ( X 1 , , X n ) be a random sample from a

Let $(X_1,\dots ,X_n)$ be a random sample from a Poisson population distribution. Derive the limiting distribution of

$T=\frac{({\overline{X}}_n-\mu )}{{\left(S_n^2/n\right)}^{1/2}}$

where $S_n^2=n^{-1}{\sum }_{i=1}^n{(X_i-{\overline{X}}_n)}^2$.