3.6.2 The Mean and Variance of X

Since $b\left(x;n,p\right)\to p\left(x;\mu \right)$ as $n\to \infty$, $p\to 0$, $np\to \mu$ , the mean and variance of a binomial variable should approach those of a Poisson variable. These limits are $np\to \mu$ and $np\left(1-p\right)\to \mu .$

Proposition

If $X$ has a Poisson distribution with parameter $\mu$ , then $E\left(X\right)=V\left(X\right)=\mu .$

These results can also be derived directly from the definitions of mean and variance.

EX 3.40 (EX 3.38 continued)