Related papers: Optimizing stakes in simultaneous bets

There are $n$ independent Bernoulli random variables $I_{k}$ with parameters $p_{k}$ that are observed sequentially. We consider a generalization of the Last-Success-Problem considering $w_{k}$ positive payments if the player successfully…

We prove the Shepp--Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof is a refinement of an argument previously…

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $…

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several…

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a…

Let $S$ be a subset of $\mathbb{R}^d$ with finite positive Lebesgue measure. The Beer index of convexity $\operatorname{b}(S)$ of $S$ is the probability that two points of $S$ chosen uniformly independently at random see each other in $S$.…

We shall show in this paper that there are experiments which are Bernoulli trials with success probability p > 0.5, and which have the curious feature that it is possible to correctly predict the outcome with probability > p.

We consider the Last-Success-Problem with $n$ independent Bernoulli random variables with parameters $p_i>0$. We improve the lower bound provided by F.T. Bruss for the probability of winning and provide an alternative proof to the one given…

This work investigates the optimal selection of the $m$th last success in a sequence of $n$ independent Bernoulli trials. We propose a threshold strategy that is $\varepsilon$-optimal under minimal assumptions about the monotonicity of the…

There are $n$ independent Bernoulli random variables with parameters $p_i$ that are observed sequentially. Two players, A and B, act in turns starting with player A. Each player has the possibility on his turn, when $I_k=1$, to choose…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

Suppose $N$ independent Bernoulli trials are observed sequentially at random times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. We focus…

In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\sigma_n \mathbb{P}(S_n\!=\!j)\leq\eta$ where $\sigma_n$ denotes the…

Consider the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter $p$ and finite time horizon $n$. Allaart \cite{Allaart} proved that the optimal strategy…

We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion $\theta$. Assuming that an i.i.d. random sequence of Benoulli($\theta$) trials is observed sequentially, we are interested in…

We consider stochastic variational inequalities with monotone operators defined as the expected value of a random operator. We assume the feasible set is the intersection of a large family of convex sets. We propose a method that combines…

In this paper, we consider the maximization of a probability $\mathbb{P}\{ \zeta \mid \zeta \in \mathbf{K}(\mathbf x)\}$ over a closed and convex set $\mathcal X$, a special case of the chance-constrained optimization problem. We define…

Bruss's odds theorem \cite{Bruss1} addresses the problem of determining the optimal stopping time for sequences of independent indicator functions. In this note, we derive upper and lower bounds for the success probability under the optimal…

One-sided confidence intervals are presented for the average of non-identical Bernoulli parameters. These confidence intervals are expressed as analytical functions of the total number of Bernoulli games won, the number of rounds and the…

For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size…