An extremal problem related to probability
The paper considers the following problem: There is a walk from a state \(A_ 1\) to a state \(A_{n+1}\) in which the probability of remaining at \(A_ i\) is \(p_ i\), and the probability of progressing from \(A_ i\) to \(A_{i+1}\) is \(1-p_ i\). The probability \(W_{nk}\) of reaching \(A_{n+1}\) from \(A_ 1\) in exactly \(n+k\) steps which is a polynomial in the n variables \(p_ 1,...,p_ n\) is investigated: Starting with the equality \(({\mathbb{Z}}_+:=\{0,1,...\}):\) \[ W_{nk}:=W_ k(p_ 1,...,p_ n)=\prod^{n}_{i=1}(1-p_ i)\cdot \sum_{j_ 1+...+j_ n=k,j_ i\in {\mathbb{Z}}_+}p_ 1^{j_ 1}...p_ n^{j_ n}, \] the paper investigates the problem of maximizing \[ W_ k(p_ 1,...,p_ n)\quad for\quad (p_ 1,...,p_ n)\in [0,1]^ n \] and the determination of the points \((p_ 1,...,p_ n)\) for which this probability takes its maximum. The primary result of the paper is given by the following theorem: \[ W_ k(p_ 1,...,p_ n)\leq W_ k(t_ 0,...,t_ 0)=\left( \begin{matrix} n+k-1\\ k\end{matrix} \right)n^ nk^ k/(n+k)^{n+k}=:M_{nk} \] for \(n\in {\mathbb{Z}}_{++}\), \(k\in {\mathbb{Z}}_+\), \(p_ i\in [0,1]\) \((i=1,2,...,n)\), \(t_ 0:=k/(n+k)\) with equality if and only if \((p_ 1,...,p_ n)=(t_ 0,...,t_ 0)\).
scientific article; zbMATH DE number 4028560