Central limit theorem

• Is the Sample Mean of a Poisson Distribution Normal?
  For iid Poisson observations, the sample mean is a scaled Poisson variable, not exactly normal; the CLT gives its large-sample normal approximation.
• How Does a Poisson Distribution Converge to Normal?
  As a Poisson mean λ grows, centering by λ and scaling by √λ gives a standard normal limit; use a continuity correction for count probabilities.