Univariate Distribution

< List of probability distributions

In statistics, a univariate distribution represents the probability distribution of a single random variable, while a multivariate distribution represents the probability distribution of a random vector consisting of multiple random variables.

For example, the probability mass function (PMF) for the binomial distribution results in a univariate distribution because only one random variable is given in the formula:

Where n is the number of items, p is the probability and x is the random variable.

On the other hand, the multivariate gamma distribution has several variables in its probability density function (PDF):

multivariate gamma function pdf

Where Γpα is the multivariate gamma function — an extension of the gamma function for multiple variables.

Types of univariate distribution

Discrete univariate distributions differ from continuous univariate distributions in that they have a finite or countable number of values available, while the latter can take any value within a particular range or interval.

  • In a discrete distribution, a probability mass function (PMF) represents the likelihood of each value. The PMF yields the probability of each feasible value of the random variable, and the sum of all probabilities must be equal to 1.
  • In a continuous distribution, a probability density function (PDF) indicates the likelihood of each value. The PDF specifies the probability of the random variable taking on a value within a defined range, and the area under the PDF curve between two points is equal to the probability of the random variable taking on a value within that range.

Several univariate distributions exist, and some are more prevalent than others. These probability distributions are closely connected through transformations, and it is possible to invert some of them. The transformations include distributions of order statistics, taking a mixture of random variables, and truncating random variables.

univariate distribution
Connections between some common univariate distributions [1]. Discrete probability distributions are in blue; Continuous distributions are in green. Common sampling distributions are in orange.

Common discrete univariate distributions:

Continuous univariate distributions:

References

[1] Leemis, L. & McQueston, J. Teacher’s Corner: Univariate Distribution Relationships. Retrieved February 19, 2021 from https://www.academia.edu/6823496/Univariate_Distribution_Relationships.

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