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Probability Histograms01:17

Probability Histograms

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A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.
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Probability Distributions01:32

Probability Distributions

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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson...
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Binomial Probability Distribution01:15

Binomial Probability Distribution

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A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.
The outcomes of a binomial experiment fit a binomial probability distribution. A statistical experiment can be classified as a binomial experiment if the following conditions are met:
There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
There are only two possible outcomes,...
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Poisson Probability Distribution01:09

Poisson Probability Distribution

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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
The...
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Distributions to Estimate Population Parameter01:26

Distributions to Estimate Population Parameter

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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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Probability in Statistics01:14

Probability in Statistics

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Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
An example of a simple event is a coin toss. The result of a coin toss is either a head or a tail. Here, head and tail are two simple events. These two simple events make up the sample space. Further, the probability of an event occurring falls within the range of 0 to 1. The probability of an...
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Related Experiment Video

Updated: Jan 13, 2026

A Tactile Automated Passive-Finger Stimulator TAPS
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A Bayesian ARMA Probability Density Estimator.

Jeffrey D Hart1

  • 1Department of Statistics, Texas A&M University, College Station, TX 77843, USA.

Entropy (Basel, Switzerland)
|October 28, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a Bayesian method for creating Autoregressive Moving Average (ARMA) probability density estimators. This new approach offers improved efficiency and parsimony compared to traditional Fourier series methods.

Keywords:
BICKullback–Leibler discrepancyLaplace approximationidentifiabilityimportance samplingintegrated squared errorprobability bandssingle-component Metropolis–Hastings algorithmtruncated Fourier series

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Area of Science:

  • Statistics
  • Time Series Analysis
  • Probability Theory

Background:

  • Traditional methods for probability density estimation, such as Fourier series estimators, have limitations in parsimony and efficiency.
  • Autoregressive Moving Average (ARMA) models are widely used for time series analysis.

Purpose of the Study:

  • To propose a novel Bayesian approach for constructing ARMA probability density estimators.
  • To demonstrate the advantages of these Bayesian estimators over existing methods, particularly Fourier series estimators.

Main Methods:

  • A Bayesian framework is employed for estimator construction.
  • Markov Chain Monte Carlo (MCMC) methods are utilized to implement the Bayesian approach.
  • The proposed estimators are characterized as ratios of trigonometric polynomials.

Main Results:

  • The Bayesian ARMA estimators exhibit greater parsimony and efficiency under common conditions.
  • MCMC output facilitates the calculation of probability intervals for parameters and the underlying density.
  • Simulation studies assess finite sample efficiency and smoothing parameter selection.

Conclusions:

  • The proposed Bayesian approach provides a robust and efficient method for ARMA probability density estimation.
  • These estimators offer practical advantages for analyzing time series data, as illustrated with a wine attribute dataset.