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So i read over the solution presented by Andre Nicolas:

Finding mode in Binomial distribution

But i have a few questions about the whole thing:

1) why did he set the ratio as $\frac{a_{k+1}}{a_k}$? Is each possible probability value a part of a sequence?

2) what is special about setting our ratios $\geq$ and $\leq$ 1?

3) Is there something special about being an integer? Does it have to do with being a discrete distribution?

4) what is the meaning of $k = np+p-1$ vs the other scenarios where it is not equal to k?

Sorry about the flurry of questions, but what appears as me understanding what is fully going on is not the case and there are some subtle points i am missing

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D.C. the III
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  • We have $a_{k+1}\gt a_k$ if and only if $\frac{a_{k+1}}{a_k}\gt 1$, with similar results for $\ge$, $=$, $\lt$, and $\le$. The ratio of consecutive binomial coefficients is reasonably easy to compute. – André Nicolas Sep 02 '15 at 00:55
  • So are "binomial coefficients" the probabilities that we obtain if we calculated some values in the mass function? – D.C. the III Sep 02 '15 at 00:57
  • Not quite, the binomial distribution probabilities involve, in addition to the binomial coefficients $\binom{n}{j}$, powers of $p$ and $1-p$, which is often called $q$. What I found was the ratio of consecutive binomial probabilities, which involved ratios of binomial coefficients and a little more. – André Nicolas Sep 02 '15 at 01:00
  • I don't mean to be a bother, but what is the difference between the two? And also question 3 and 4 that i had. – D.C. the III Sep 02 '15 at 01:04
  • The probability $a_k$ that the binomial random variable takes on value $k$ is $\binom{n}{k}p^kq^{n-k}$. The binomial coefficient is plain $\binom{n}{k}$. I will try to answer your other question later. – André Nicolas Sep 02 '15 at 01:15
  • About $k=np+p-1$, and in particular $np+p-1$ being an integer, this has nothing to do with discreteness. The binomial distribution is a discrete distribution for all values of $n$ and $p$. The condition $k=np+p-1$ comes out of the analysis, it is the condition for bimodality. – André Nicolas Sep 02 '15 at 01:20
  • You are welcome. I hope others will provide extended answers. – André Nicolas Sep 02 '15 at 01:21
  • Then it is the analysis of what's happening that is confusing me – D.C. the III Sep 02 '15 at 01:22

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