Maximum of a binomial distribution

Question

I was reading J.W. Rohlf's "Modern Physics from α to Z^0" and, on appendix D, when talking about the binomial distribution in the limit where n (the number of trials) is very large, he says (without proof) that the distribution has a maximum at the average value np.

How can I prove that the maximum of a binomial distribution when $n$ is very large lies at the average value $np$?

Duplicate of https://math.stackexchange.com/questions/117926/finding-mode-in-binomial-distribution?noredirect=1&lq=1 — heropup, Dec 30 '19 at 19:41
I added a paragraph at the beginning for motivation and context. — Dirac, Dec 31 '19 at 17:35

score 1 · Answer 1 · answered Dec 30 '19 at 18:51

1

It doesn't, if $np$ is not an integer.

answered Dec 30 '19 at 18:51

Robert Israel

470,583

score 0 · Answer 2 · answered Dec 30 '19 at 19:36

HINT: one way is to compare the ratio

$${P(Bin(n,p) = k+1) \over P(Bin(n,p) = k)}$$

and try to notice different trends for different regimes of $k$. When done carefully, this should give you the answer when $np$ is not an integer.

Full solution: available in wikipedia

Maximum of a binomial distribution

2 Answers2