## Normal Approximation to Binomial Distribution

Let $X$ be a Poisson distributed random variable with mean $\lambda$.

The mean of $X$ is $\mu=E(X) = \lambda$ and variance of $X$ is $\sigma^2=V(X)=\lambda$.

The general rule of thumb to use normal approximation to Poisson distribution is that $\lambda$ is sufficiently large (i.e., $\lambda \geq 5$).

For sufficiently large $\lambda$, $X\sim N(\mu, \sigma^2)$. That is $Z=\frac{X-\mu}{\sigma}=\frac{X-\lambda}{\sqrt{\lambda}} \sim N(0,1)$.

## Formula

• $P(X=A)=P(A-0.5<X<A+0.5)$
• $P(X<A)=P(X<A-0.5)$
• $P(X\leq A)=P(X<A+0.5)$
• $P(A< X\leq B)=P(A-0.5<X<B+0.5)$
• $P(A\leq X< B)=P(A-0.5<X<B-0.5)$
• $P(A\leq X\leq B)=P(A-0.5<X<B+0.5)$