Mean, variance and Poisson approximation to the binomialEdexcel A-Level Further Maths: Revision notes

If $X\sim B(n,p)$, with $n$ independent trials each with success probability $p$, then $E(X)=np,\qquad\text{Var}(X)=np(1-p).$ Derivations are not required. Example: $X\sim B(40,0.15)$ has $E(X)=6$ and $\text{Var}(X)=40(0.15)(0.85)=5.1$. The variance is always smaller than the mean, because it is the mean multiplied by $1-p$. The standard deviation is the square root of the variance, here $2.26$.

If $Y\sim\text{Po}(\lambda)$ then $E(Y)=\lambda,\qquad\text{Var}(Y)=\lambda.$ The mean and variance are equal. Derivations are not required. If $Y\sim\text{Po}(7)$ the variance is $7$ and the standard deviation is $\sqrt7=2.65$. This equality is a useful check: if a data set has a mean and variance that are very different, a Poisson model is probably unsuitable. In contrast, a binomial has variance less than its mean.

If you know $E(X)$ and $\text{Var}(X)$ for a binomial variable, divide to find $1-p$: $\frac{\text{Var}(X)}{E(X)}=\frac{np(1-p)}{np}=1-p$. Then find $n$ from $n=\frac{E(X)}{p}$. Example: mean $12$ and variance $9.6$. Then $1-p=\frac{9.6}{12}=0.8$, so $p=0.2$ and $n=\frac{12}{0.2}=60$. Check: $60(0.2)=12$ and $60(0.2)(0.8)=9.6$. Always check that $p$ lies between 0 and 1 and that $n$ is a positive integer.

When $n$ is large and $p$ is small, $B(n,p)$ can be approximated by $\text{Po}(np)$. Derivations are not required. The approximation works because $np(1-p)\approx np$ when $p$ is small, so the binomial mean and variance are nearly equal, as for a Poisson. As a guide, $n$ should be at least about 50 and $p$ no more than about $0.1$. Example: $X\sim B(500,0.004)$ has mean 2, so $X\approx\text{Po}(2)$. Then $P(X\geq4)\approx1-P(\text{Po}(2)\leq3)=0.1429$, compared with the exact $0.1425$.

Compare the means and variances of the two models. The approximation keeps the mean $np$ but replaces the variance $np(1-p)$ by $np$, so it overestimates the spread, by a factor of $\frac{1}{1-p}$. For $p=0.004$ the effect is tiny. For $B(20,0.3)$ the mean is 6 and the variance 4.2, while $\text{Po}(6)$ has variance 6; and $P(Y\leq3)$ is $0.107$ exactly but $0.151$ under the approximation, about 41% too large. Conclude that the approximation is unsuitable when $p$ is not small. Use evidence from the numbers and a clear conclusion.