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$f_{n,p}(k)$ is the probability mass function of a binomial distribution with parameters $n$ and $p$ i.e, for $k \in \{0,1,2, \cdots,n\}$, $f_{n,p}(k) = \binom{n}{k}p^k(1-p)^{n-k}$. Let $F_{n,p}$ be the corresponding cumulative distribution function, i.e. $F_{n, p}(k) = \sum_{i=0}^k f_{n,p}(k)$ for $k \in \{0,1,2,\cdots,n\}.$ Consider the function $h_n(p)$ defined for $0\leq p \leq 1$ as follows: \begin{equation} h_n(p) = \sum_{k=1}^n f_{n,p}(k)F_{n,p}(k-1). \end{equation} After plotting the function for different values of $n,$ I think that the following two properties hold for $h_n(p):$

  1. $h_n(p) \leq h_{n+1}(p)$ for all $0\leq p \leq 1$ and $n= 1,2,3, ...\, .$

  2. $h_n(p)$ is a concave function for $n= 1,2,3, ...\,.$

Am trying to prove the above two statements analytically. I would really appreciate any help with the proofs.

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  • $\begingroup$ Let $g_n(p) := \sum_{k=0}^n \binom{n}{k}^2 \cdot p^{2k} \cdot (1-p)^{2(n-k)}$. Note that $h_n(p) = h_n(1-p)$ and $1 = h_n(p) + h_n(1-p) + g_n(p)$. (Write 1 as double sum.) Thus a) is equivalent to $g_n(p) \geq g_{n+1}(p)$ and b) to the convexity of $g_n$. I think I have seen this elsewhere (Marshall/Olkin ?). $\endgroup$ Jan 13, 2020 at 0:01
  • $\begingroup$ Thanks a lot for the hint. Could you please let me know which paper/book of Marshall/Olkin to look at ? $\endgroup$
    – user151031
    Jan 13, 2020 at 6:50
  • $\begingroup$ The book is: A.W. Marshall, I. Olkin: Inequalities: Theory of Majorization and its Applications,, Academic Press, 1979. A further comment (surely known to you): Let $X,Y$ be independent random variables $\sim Bin(n,p)$, then $h_n(p) = P(X<Y) = P(X>Y)$ and $P(X=Y) = g_n(p)$, $\endgroup$ Jan 13, 2020 at 10:26
  • $\begingroup$ I glanced through the entire book but could not come across something related to my question. Could you please let me know which section to look at ? I initially thought it should be straightforward to prove convexity of the function $g_n(p)$. But even that is turning out to be more difficult than I had imagined. Any hints/references please. Thanks! $\endgroup$
    – user151031
    Jan 13, 2020 at 20:38
  • $\begingroup$ Sorry, I had chapters 11 (Stochastic Majorizations) and 12 (Probabilistic and Statistical Applicatona) in mind, but the content has nothing to do with your problem. At first sight your problem seems to be simple and I still think there must be an elementary solution. I will think about it, now its of interest to me. $\endgroup$ Jan 14, 2020 at 21:44

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