< List of probability distributions < Absolutely continuous distribution
What is an absolutely continuous distribution?
In basic terms, an absolutely continuous distribution is one that can be represented by a continuous density function. The density function assigns a probability to each possible value of a random variable. Most distributions we use in statistics are absolutely continuous.
All absolutely continuous distributions are continuous, but not all continuous distributions are absolutely continuous. There are singular distributions with no density, such as the Cantor distribution, that cannot be classified as absolutely continuous, discrete or mixed [3].
The main difference in absolute continuous vs continuous distributions is in how the distribution’s function is defined:
- A continuous distribution is continuous as a function: like continuous functions, you can draw a continuous distribution on paper without lifting a pen.
- An absolutely continuous distribution has a density function with respect to a Lebesgue measure [2]. A Lebesgue measure is used in probability to assign a size to sets of real numbers; it means that the density function can be used to calculate the probability of any set of real numbers.
If a random variable has an absolutely continuous distribution, its underlying sample space must be uncountably infinite [4]. This is because the density function must be continuous, and a continuous function can take on an infinite number of values.
Absolutely continuous distribution examples
Nearly all probability distributions used in statistics, including those found in statistical software packages, are absolutely continuous. Examples include:
- Arcsine distribution
- Beta-distribution
- Cauchy distribution
- Chi-squared distribution
- Exponential distribution
- Gamma distribution
- Normal distribution
- Student’s t-distribution
- Uniform distribution.
For most analyses, the difference between a continuous and absolutely continuous function is so subtle it can usually be ignored. Where continuous vs. absolutely continuous needs closer scrutiny is when you’re working within measure theory.
Definition of absolutely continuous
Absolute continuity is a concept used in measure theory, where one measure is “absolutely” determined by another. To put this another way, if one measure assigns zero probability to a set, then the other measure must also assign zero probability to that set. We can also state this a little more formally:
Suppose that two measures, μF and λ are defined on the same σ-algebra. Measure μF is absolutely continuous with respect to λ if, for every measurable set E, λ(E) = 0 implies μF(E) = 0. In other words, relative to λ, there are no measure zero sets to which μF assigns nonzero probability. This means that if a set has zero measure under λ, then it must also have zero measure under μF.
This concept allows us to compare different measures and infer their relationship. For example, if we know that one measure is absolutely continuous with respect to another, then we know that the two measures cannot assign different probabilities to the same set.
References
- MArkSweep via Wikimedia Commons. Licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- I. Ahmad and Pi-Erh Lin, “A nonparametric estimation of the entropy for absolutely continuous distributions (Corresp.),” in IEEE Transactions on Information Theory, vol. 22, no. 3, pp. 372-375, May 1976, doi: 10.1109/TIT.1976.1055550.
- Probability distribution. Retrieved August 8, 2023 from: https://academic-accelerator.com/encyclopedia/probability-distribution
- Healy, P. Lecture 6: Expectation is a positive linear operator.