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 $x$?
2 Answers
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Hint...try simplifying $$\frac{p(X=r+1)}{p(X=r)}$$
answered Apr 6, 2016 at 20:53
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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.
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$\begingroup$ How does this lead to the value for maximum? $\endgroup$ Commented Apr 6, 2016 at 20:59
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$\begingroup$ @user329265: all you need is in this answer. $\endgroup$– user65203Commented Apr 6, 2016 at 21:00
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$\begingroup$ Ok. I'll try to work on it. I'll let you know if I have any questions. $\endgroup$ Commented Apr 6, 2016 at 21:01