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Related Concept Videos

Poisson Probability Distribution01:09

Poisson Probability Distribution

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...
Expected Value01:15

Expected Value

The expected value is known as the "long-term" average or mean. This means that over the long term of experimenting over and over, you would expect this average. The expected average is represented by the symbol μ. It is calculated as follows:In the equation, x is an event, and P(x) is the probability of the event occurring.The expected value has practical applications in decision theory.This text is adapted from Openstax, Introductory Statistics, Section 4.2 Mean or Expected Value and...
Expected Frequencies in Goodness-of-Fit Tests01:19

Expected Frequencies in Goodness-of-Fit Tests

A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the dataset. Suppose the expected frequencies for a dataset are equal such as when predicting the frequency of any number appearing when casting a die. In that case, the expected frequency is the ratio of the total number of observations (n) to the number of categories (k).
Probability Distributions01:32

Probability Distributions

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 probability...
Determination of Expected Frequency01:08

Determination of Expected Frequency

Suppose one wants to test independence between the two variables of a contingency table. The values in the table constitute the observed frequencies of the dataset. But how does one determine the expected frequency of the dataset? One of the important assumptions is that the two variables are independent, which means the variables do not influence each other. For independent variables, the statistical probability of any event involving both variables is calculated by multiplying the individual...
Binomial Probability Distribution01:15

Binomial Probability Distribution

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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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Published on: July 3, 2020

Expectation values for integer powers of a Poisson-distributed random number.

Julian Henn1

  • 1Laboratory of Crystallography, Universität Bayreuth, 95440 Bayreuth, Germany. julian.henn@uni-bayreuth.de

Acta Crystallographica. Section A, Foundations of Crystallography
|October 19, 2012
PubMed
Summary
This summary is machine-generated.

This study demonstrates that expectation values for Poisson-distributed random numbers extend to negative integer powers, not just positive ones. A new recursion formula simplifies calculating these powers and yields analytical expressions using hypergeometric functions.

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Area of Science:

  • Probability Theory
  • Statistical Mechanics
  • Number Theory

Background:

  • Expectation values are fundamental in probability theory.
  • Poisson distributions are widely used in various scientific fields.
  • Powers of random variables are essential for statistical analysis.

Purpose of the Study:

  • To investigate the existence and calculation of expectation values for negative integer powers of Poisson-distributed random numbers.
  • To develop a generalized method for computing these expectation values.
  • To establish an analytical representation for these values.

Main Methods:

  • Derivation of a recursion formula for expectation values of powers.
  • Application of the recursion formula to positive and negative integer powers.
  • Utilizing hypergeometric functions for analytical representation.

Main Results:

  • Established the existence of expectation values for negative integer powers of Poisson-distributed random numbers.
  • Developed a novel recursion formula relating expectation values of consecutive powers.
  • Provided an analytical solution for expectation values of integer powers using hypergeometric functions.

Conclusions:

  • The study expands the understanding of expectation values for Poisson distributions.
  • The derived recursion formula offers an efficient computational tool.
  • The analytical representation using hypergeometric functions provides deeper theoretical insights.