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About 6,397 papers for “Probability mass function”

Ordered Probability Mass Function

Zhengjun Cao, Lihua Liu · arXiv (Cornell University) · 2013

Suppose that in the four tests Alice's scores are 90, 95, 85, 90, and Bob's scores are 85, 95, 90, 90. How to evaluate their scores? In this paper, we introduce the concept of ordered probability mass function which can be used to find a probability mass function with smaller variance. More interestingly, we can use it to distinguish sequences of positive numbers statistically.

Cited by 0Open access

A Conditional Probability Mass Function

Wolfram Demonstrations Project · 2008

The probability mass function of a pair of discrete random variables is the function The conditional mass function of given is the function Thus the mass function lefthand plot computes probabilities of intersections while the conditional mass function righthand plot computes conditional probabilities For each value the slice through the conditional mass function at that value gives the distribu

Cited by 0Open access

A Conditional Probability Mass Function

Wolfram Demonstrations Project · 2008

The probability mass function of a pair of discrete random variables is the function The conditional mass function of given is the function Thus the mass function lefthand plot computes probabilities of intersections while the conditional mass function righthand plot computes conditional probabilities For each value the slice through the conditional mass function at that value gives the distribu

Cited by 1Open access

A conditional probability mass function

Boucher, Chris · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2013

The probability mass function of a pair of discrete random variables (X,Y) is the function f(x,y)=P(X=x,Y=y). The conditional mass function of Y given X is the function f(y|x)=P(Y=y|X=x). Thus the mass function (left-hand plot) computes probabilities of intersections, while the conditional mass function (right-hand plot) computes conditional probabilities. For each x value, the slice through the conditional mass function at that value gives the distribution of Y when X assumes the value x. The mean of this distribution is the conditional expectation of Y given X=x, E[Y|X=x]. The weighted average of the conditional expectations, with the weights given by the probability that X=x, is the expected value of Y.You can reverse the roles of X and Y to obtain the conditional mass function of X given Y and the conditional expectation of X given Y=y

Cited by 0

Exact Binomial Probability Mass Function Simplified

Ilija Barukčić · Zenodo (CERN European Organization for Nuclear Research) · 2026

Background:The binomial distribution is fundamental in probability theory and statistics. While approximations such as the Poisson distribution are useful, an exact algebraic simplification can be advantageous. Material and methods:For large sample sizes \(n\), computational precision issues may arise even when using the exact binomial formula. Results:An alternative exact representation of the binomial PMF using the substitution \(x = n - k\) and Pochhammer symbol or the rising factorial (x), leading to a form that separates the factorial ratio into a product and highlights the role of the complement probability is derived. The resulting expression is\begin{align}\boxed{P(X = k) = \frac{(k+1)^{(x)}}{x!} \cdot (1-p)^{x} \cdot e^{k \ln p}}\end{align}Conclusion:The simplified exact formulation of the binomial probability mass function offer various computational and other benefits in certain settings.

Cited by 0Open access

Exact Binomial Probability Mass Function Simplified

Ilija Barukčić · Zenodo (CERN European Organization for Nuclear Research) · 2026

Background:The binomial distribution is fundamental in probability theory and statistics. While approximations such as the Poisson distribution are useful, an exact algebraic simplification can be advantageous. Material and methods:For large sample sizes \(n\), computational precision issues may arise even when using the exact binomial formula. Results:An alternative exact representation of the binomial PMF using the substitution \(x = n - k\) and Pochhammer symbol or the rising factorial (x), leading to a form that separates the factorial ratio into a product and highlights the role of the complement probability is derived. The resulting expression is\begin{align}\boxed{P(X = k) = \frac{(k+1)^{(x)}}{x!} \cdot (1-p)^{x} \cdot e^{k \ln p}}\end{align}Conclusion:The simplified exact formulation of the binomial probability mass function offer various computational and other benefits in certain settings.

Cited by 0Open access

Estimating a probability mass function with unknown labels

Anevski, Dragi, Gill, Richard D., Zohren, Stefan · arXiv (Cornell University) · 2013

In the context of a species sampling problem we discuss a non-parametric maximum likelihood estimator for the underlying probability mass function. The estimator is known in the computer science literature as the high profile estimator. We prove strong consistency and derive the rates of convergence, for an extended model version of the estimator. We also study a sieved estimator for which similar consistency results are derived. Numerical computation of the sieved estimator is of great interest for practical problems, such as forensic DNA analysis, and we present a computational algorithm based on the stochastic approximation of the expectation maximisation algorithm. As an interesting byproduct of the numerical analyses we introduce an algorithm for bounded isotonic regression for which we also prove convergence.

Cited by 0Open access

A snapshot of the calculated probability mass function.

Garrett Jenkinson (167097), John Goutsias (167102) · Figshare · 2015

<p>Joint conditional probability mass function of susceptible and infected pupils at the end of the 6th day of the influenza epidemic.</p>

Cited by 0Open access

Estimating a probability mass function with unknown labels

Dragi Anevski, Richard D. Gill, Stefan Zohren · The Annals of Statistics · 2017

In the context of a species sampling problem, we discuss a nonparametric maximum likelihood estimator for the underlying probability mass function. The estimator is known in the computer science literature as the high profile estimator. We prove strong consistency and derive the rates of convergence, for an extended model version of the estimator. We also study a sieved estimator for which similar consistency results are derived. Numerical computation of the sieved estimator is of great interest for practical problems, such as forensic DNA analysis, and we present a computational algorithm based on the stochastic approximation of the expectation maximisation algorithm. As an interesting byproduct of the numerical analyses, we introduce an algorithm for bounded isotonic regression for which we also prove convergence.

Cited by 15Open access

Maximum Likelihood Set for Estimating a Probability Mass Function

Bruno Jedynak, Sanjeev Khudanpur · Neural Computation · 2005

We propose a new method for estimating the probability mass function (pmf) of a discrete and finite random variable from a small sample. We focus on the observed counts--the number of times each value appears in the sample--and define the maximum likelihood set (MLS) as the set of pmfs that put more mass on the observed counts than on any other set of counts possible for the same sample size. We characterize the MLS in detail in this article. We show that the MLS is a diamond-shaped subset of the probability simplex [0,1]k bounded by at most k x (k-1) hyper-planes, where k is the number of possible values of the random variable. The MLS always contains the empirical distribution, as well as a family of Bayesian estimators based on a Dirichlet prior, particularly the well-known Laplace estimator. We propose to select from the MLS the pmf that is closest to a fixed pmf that encodes prior knowledge. When using Kullback-Leibler distance for this selection, the optimization problem comprises finding the minimum of a convex function over a domain defined by linear inequalities, for which standard numerical procedures are available. We apply this estimate to language modeling using Zipf's law to encode prior knowledge and show that this method permits obtaining state-of-the-art results while being conceptually simpler than most competing methods.

Cited by 27

An Information Theoretic Approach to Probability Mass Function Truncation

Ferdinando Cicalese, Luisa Gargano, Ugo Vaccaro · 2019

Given a discrete random variable X that takes values in a finite set χ according to a probability mass function (pmf) P, a truncated pmf Q of P is a conditional pmf that results from restricting the domain of X to some subset of χ. Truncated pmf arise in several problems of statistics and probability. In this paper, we propose and analyze a few criteria to truncate pmf's so that the truncated one is as much close as possible to the original pmf, under different information theoretic measures of distance.

Cited by 2

Efficient Probability Mass Function Estimation from Partially Observed Data

Joseph K. Chege, Mikus J. Grasis, Alla Manina et al. · 2022

Estimating the joint probability mass function (PMF) of a set of random variables from partially observed data is a crucial part of statistical learning and data analysis, with applications in areas such as recommender systems and data classification. Recently, it has been proposed to estimate the joint PMF based on the maximum likelihood (ML) of the data, fitted to a low-rank canonical polyadic decomposition (CPD) model of the joint PMF. To this end, a hybrid alternating-directions expectation-maximization (AD-EM) algorithm was proposed to solve the ML optimization problem, consisting of computationally expensive AD iterations followed by an EM refinement stage. It is well known that the convergence rate of EM decreases as the fraction of missing data increases. In this paper, we address the slow convergence of the EM algorithm. By adapting the squared iterative methods (SQUAREM) acceleration scheme to the context of PMF estimation, we propose the SQUAREM-PMF algorithm to speed up the convergence of the EM algorithm. Moreover, we demonstrate that running the computationally cheaper EM algorithm alone after an appropriate initialization is sufficient. Numerical results on both synthetic and real data in the context of movie recommendation show that our algorithm outperforms state-of-the-art PMF estimation algorithms.

Cited by 5

Recursive Probability Mass Function Method to Calculate Probability Distributions of Pulse-Shaped Signals

Tomoya Fukami, Hirobumi Saito, Akira Hirose · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2023

This paper proposes an accurate and efficient method to calculate probability distributions of pulse-shaped complex signals. We show that the distribution over the in-phase and quadrature-phase (I/Q) complex plane is obtained by a recursive probability mass function of the accumulator for a pulse-shaping filter. In contrast to existing analytical methods, the proposed method provides complex-plane distributions in addition to instantaneous power distributions. Since digital signal processing generally deals with complex amplitude rather than power, the complex-plane distributions are more useful when considering digital signal processing. In addition, our approach is free from the derivation of signal-dependent functions. This fact results in its easy application to arbitrary constellations and pulse-shaping filters like Monte Carlo simulations. Since the proposed method works without numerical integrals and calculations of transcendental functions, the accuracy degradation caused by floating-point arithmetic is inherently reduced. Even though our method is faster than Monte Carlo simulations, the obtained distributions are more accurate. These features of the proposed method realize a novel framework for evaluating the characteristics of pulse-shaped signals, leading to new modulation, predistortion and peak-to-average power ratio (PAPR) reduction schemes.

Cited by 1Open access

Spectral Enhancement of Whispered Speech Based on Probability Mass Function

Hamid Sharifzadeh, Ian McLoughlin, Farzaneh Ahmadi · 2010

Whispered speech can be effectively used for quiet and private communications over mobile phones and is also the communication means for ENT patients under a regime of voice rest. The reconstruction of natural sounding speech from such whispers can be useful for several types of application across different scientific fields ranging from communications to biomedical engineering. Despite the useful applications for a such technology, the reconstruction of natural speech from whispers has received relatively little research effort to date. This paper presents novel methods for spectral enhancement and formant smoothing with the aim of attaining more natural sounding speech within the reconstruction process. The proposed approach uses a probability mass-density function to identify a reliable formant trajectory through whispers and apply vocal modifications accordingly. Subjective evaluation experiments were performed, and are reported, to assess the performance of the techniques. A method for the near real-time conversion of whispers to normal phonated speech through a modified CELP codec has been discussed in our previously published work which, the proposed formant modification approach in this paper builds upon.

Cited by 4