The Normal Approximation of the Poisson Distribution

If $X \sim \text{Poisson}(\lambda),$ then $E[X] = \lambda$ and $\text{Var}(X) = \lambda.$ For large $\lambda$ (a common rule of thumb is $\lambda \geq 10$), the Poisson distribution is well-approximated by a normal distribution with the same mean and variance:

$$X \sim \text{Poisson}(\lambda) \;\;\stackrel{\text{approx}}{\sim}\;\; N(\lambda,\,\lambda).$$

In particular, the standard deviation of the approximating normal is $\sigma = \sqrt{\lambda}.$

Since Poisson is discrete but the normal is continuous, we apply a continuity correction: shift each boundary by $0.5$ in the direction that includes more of the distribution. For a left tail $P(X \leq k),$ we shift up to $k + 0.5{:}$

$$P(X \leq k) \approx P(Y \leq k + 0.5) = P\!\left(Z \leq \dfrac{k + 0.5 - \lambda}{\sqrt{\lambda}}\right).$$

For example, if $X \sim \text{Poisson}(25),$ then $\mu = 25$ and $\sigma = 5,$ so

$$P(X \leq 30) \approx P\!\left(Z \leq \dfrac{30.5 - 25}{5}\right) = P(Z \leq 1.10) \approx 0.8643.$$