Binomial Distribution at Infinity

Probability Level 1

Consider a binomial distribution of $X\sim B(n,p)$ .

It can be easily shown that $P(X=k)={n\choose k}p^k{(1-p)}^{n-k}$ for $k=0,1,2,3,\ldots,n$ .

Now, let's take the limit of the above using $n \to \infty$ . Instead of having an infinitesimal $p$ , let's assume that it is given that $np$ , the mean of the probability distribution function, is some finite value $m$ .

Find $P(X=k)$ in terms of $m$ and $k$ for this new distribution, where $k=0,1,2,3,\ldots$ , without looking anything up or reciting any formulas from memory.

0

\frac{e^{-m}}{k!\ m}

\frac{e^{-k}}{m!\ k}

\frac{m^ke^{-m}}{k!}

\frac{m}{k! \ln{m}}

\frac{\ln{k}}{k! \ln{m}}

\frac{e^{-k}\ln{m}}{m!\ k}

Limit does not exist