Showing $ \mathbb{P}\Big( \frac{\Pi - \lambda}{\sqrt{\lambda}}\leq x \Big) $

Question

Let $\Pi$ be a random variable distributed by Poisson distribution with parameter $\lambda>0.$ Need to show that $$ \mathbb{P}\Big( \frac{\Pi - \lambda}{\sqrt{\lambda}}\leq x \Big) \rightarrow_{\lambda \to \infty} \Phi(x)$$ for every $x \in \mathbb{R}.$

I have no idea what to do and how to start. I know that Poisson random variable $exp(iat + \lambda(e^{ibt-1})$ maybe this is equal to $\Phi(t)$ but than what is $a$ and $b$?

Answer 1

Hint: Use the central limit theorem and the fact that exponentials transform sums in products.