I need help solving part b of this question:
The random variable $X \sim ~Po(m)$.
Given that $P(X=k+1)=P(X=k-1)$, where $k$ is a positive integer,
a) show that $m^2=k(k+1)$
b) hence show that the mode of $X$ is $k$.
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2This seems like kind of a roundabout way to do it to me. The main idea is that for $X \sim Po(\lambda)$, $\frac{P(X=k)}{P(X=k+1)}=\frac{k+1}{\lambda}$. One mode is the smallest $k$ such that this is greater than $1$. (It is possible that $k-1$ is another mode, if $\lambda$ is an integer.) – Ian Apr 07 '17 at 05:20
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Hi, thanks for your answer, but I don't quite understand how you arrive at $ \frac{P(X=k)}{P(X=k+1)}= \frac{k+1}{λ}$ and also what you meant by the sentence after that. – The 'Fe' Man Apr 07 '17 at 05:28
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The equation is just algebra with the definition of the Poisson PMF, taking advantage of rules of exponentials and factorials. The following sentence is saying that there is a mode (i.e. a maximizer of $P(X=k)$) at the smallest $k$ such that $P(X=k)/P(X=k+1)>1$. Try to see why this is the case. (It might help to graph the PMF for, say, $\lambda=6.5$ to get a feel for the picture.) – Ian Apr 07 '17 at 05:40