Let $\displaystyle Y_n $ be a sequence of Poisson distributed random variables with mean n*u. Let $\displaystyle X_n = {Y_n}^{1/2} $. Find the limiting distribution of $\displaystyle X_n $ after properly standardized.

I'm pretty sure that X should be normally distributed and the answer should use the CLT in some way. All I have so far is that:

$\displaystyle P(X=x) = \frac{e^{-nu}(nu)^{x^2}}{{x^2}!} $

And now i'm pretty much stuck