I have this question.
When modelling counts Y defined on ${0,1,2,...} $ , it is common to use the Poisson distribution, when $ E[Y] = var[Y] $ is implied. Show that the negative binomial distribution may be a better choice if $var[Y] > E[Y]$.
The definitions of the NBIN are
$ \sigma^2=Var(x)=\dfrac{r(1-p)}{p^2} $
$ \mu=E(X)=\dfrac{r}{p} $
The problem is that for values of $ p > 0.5 $, then the $var[Y] < E[Y]$, not $>$ as the question is suggesting.
How does the above statement hold and how am I meant to prove this?