Questions on Binomial Distribution

Question

Given that $X$ follows a binomial distribution and has parameters $n,p$, how can we prove that as $x$ goes from $0$ to $n$, $P(X=x)$ increases monotonically at first but when it reaches its maximum, it decreases monotonically? Also, what is that maximum $x$?

Answer 1

Hint...try simplifying $$\frac{p(X=r+1)}{p(X=r)}$$

Answer 2

The law is expressed by

$$P_k=\binom nkp^kq^{n-k},$$ where $q:=1-p$.

The trick is to take the ratio of two successive values,

$$\frac{P_{k+1}}{P_k}=\frac{n-k}{k+1}\frac pq$$ and compare it to $1$.

When $(n-k)p>(k+1)q$ the terms go increasing, and conversely. This relation can be rewritten

$$k<np-q.$$

You should be able to conclude.