Normal approximation to the binomial and PoissonEdexcel International A Level Further Maths: Revision notes

Binomial and Poisson probabilities are tedious to add up when $n$ or $\lambda$ is large. A normal approximation replaces the discrete distribution with a continuous one that has the same mean and variance. For $X\sim\mathrm{B}(n,p)$ the approximation is good when $n$ is large and $p$ is close to $0.5$ (so the distribution is nearly symmetrical). A usual check is $np>5$ and $nq>5$. For $X\sim\mathrm{Po}(\lambda)$ the approximation is good when $\lambda$ is large, usually $\lambda>10$.

If $X\sim\mathrm{B}(n,p)$ with $n$ large and $p$ near $0.5$, then $X\approx\mathrm{N}(np,\ np(1-p)).$ The mean is $np$ and the variance is $npq$ where $q=1-p$. The second parameter of $\mathrm{N}(\mu,\sigma^2)$ is the variance, so standardise with $\sigma=\sqrt{npq}$. Example: $X\sim\mathrm{B}(80,0.4)$ gives $\mu=32$, $\sigma^2=19.2$, so $X\approx\mathrm{N}(32,19.2)$.

If $X\sim\mathrm{Po}(\lambda)$ with $\lambda$ large, then $X\approx\mathrm{N}(\lambda,\ \lambda).$ A Poisson distribution has mean and variance both equal to $\lambda$, so the variance is $\lambda$ and the standard deviation is $\sqrt\lambda$. Example: $X\sim\mathrm{Po}(25)$ gives $X\approx\mathrm{N}(25,25)$ with $\sigma=5$.

$X$ takes whole-number values but the normal variable $Y$ is continuous. Each integer $k$ is treated as the block $k-0.5<Y<k+0.5$. This adjustment is the continuity correction. $\mathrm{P}(X=k)\approx\mathrm{P}(k-0.5<Y<k+0.5)$ $\mathrm{P}(X\le k)\approx\mathrm{P}(Y<k+0.5)$ and $\mathrm{P}(X<k)\approx\mathrm{P}(Y<k-0.5)$ $\mathrm{P}(X\ge k)\approx\mathrm{P}(Y>k-0.5)$ and $\mathrm{P}(X>k)\approx\mathrm{P}(Y>k+0.5)$ Rewrite strict inequalities as inclusive ones first: $X<20$ means $X\le19$, so the boundary is $19.5$.

$X\sim\mathrm{Po}(25)$. Find $\mathrm{P}(X\ge30)$. Approximate by $Y\sim\mathrm{N}(25,25)$. Then $\mathrm{P}(X\ge30)\approx\mathrm{P}(Y>29.5)$. $z=\frac{29.5-25}{5}=0.9$, so the probability is $1-\Phi(0.9)=1-0.8159=0.184$. The exact Poisson value is $0.182$, so the approximation is accurate to two decimal places.

$X\sim\mathrm{B}(150,0.45)$. Find $\mathrm{P}(60\le X<75)$. $X\approx\mathrm{N}(67.5,37.125)$ and $60\le X<75$ means $X=60,\dots,74$, so use $59.5<Y<74.5$. $z$ values: $\frac{59.5-67.5}{6.093}=-1.31$ and $\frac{74.5-67.5}{6.093}=1.15$. Probability $=0.8747-0.0946=0.780$. For an inverse problem, such as the smallest $k$ with $\mathrm{P}(X\le k)\ge0.95$ for $X\sim\mathrm{Po}(36)$, solve $\frac{k+0.5-36}{6}\ge1.645$ for $k$, then round up to an integer.