Continuous Probability Function

A Continuous Probability Function is a probability distribution function that can accept a continuous random variable.

• AKA: PDF, Probability Density.
• Context:
  • It has an Area of 1 (Integrable).
  • It has an Area under some Interval that is a Probability Value.
  • It can (often) be a member of a Continuous Probability Distribution Family.
  • It can be a Cumulative Density Function, if the sample space is an ordered set and one of the end points is assumed (negative infinity).
  • It can range from being an Univariate Density Function to being a Multivariate Density Function.
  • It can range from being an Abstract Continuous Probability Function to being a Continuous Probability Function Structure.
  • It can range from being a Uniform Density Function to being a Non-Uniform Density Function.
  • It can be instantiated in a Continuous Probability Density Multiset (by a Probability Density Estimation Task).
• Example(s):
  • a Gaussian Density Function(x, μ=1, σ=0.5).
  • an Exponential Density Function, from an exponential density distribution.
  • a Gamma Density Function, from a gamma density distribution.
  • a Weibull Density Function, from a Weibull density distribution.
  • a Beta Density Function, from a beta density distribution.
  • a Poisson Density Function, from a Poisson density distribution.
  • a Hyper Geometric Probability Function.
  • a Mixture Density Function.
  • a Conditional Probability Density Function.
  • …
• Counter-Example(s):
  • any Discrete Probability Function.
• See: Likelihood Function, Volume.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/probability_density_function Retrieved:2015-6-24.
  • In probability theory, a probability density function ('PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability of the random variable falling within a particular range of values is given by the integral of this variable’s density over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

    The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function".

2011

• (Wikipedia, 2011) http://en.wikipedia.org/wiki/Probability_density_function
  • QUOTE: In probability theory, a probability density function ('pdf), or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density functionis nonnegative everywhere, and its integral over the entire space is equal to one.

    The terms “probability distribution function” and “probability function” have also sometimes been used to denote the probability density function. However, special care should be taken around this usage since it is not standard among probabilists and statisticians. In other sources, “probability distribution function” may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the “probability mass function”.

• http://en.wikipedia.org/wiki/Continuous_probability_distribution#Continuous_probability_distribution
  • QUOTE: A continuous probability distribution shall be understood as a probability distribution that has a probability density function. Mathematicians also call such distribution 'absolutely continuous, since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

    Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.

2006

• (Dubnicka, 2006c) ⇒ Suzanne R. Dubnicka. (2006). “Random Variables - STAT 510: Handout 3." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • TERMINOLOGY : A random variable is said to be continuous if its support set is uncountable (i.e., the random variable can assume an uncountably infinite number of values).
  • ALTERNATE DEFINITION: A random variable is said to be continuous if its cdf FX(x) is a continuous function of x.
  • TERMINOLOGY : Let X be a continuous random variable with cdf FX(x). The probability density function (pdf) for X, denoted by fX(x), is given by fX(x) = d/dx FX(x),
  • The pmf of a discrete random variable and the pdf of a continuous random variable provides complete information about the probabilistic properties of a random variable. However, it is sometimes useful to employ summary measures. The most basic summary measure is the expectation or mean of a random variable X, denoted E(X), which can be thought of as an “average” value of a random variable.

2000

• (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
  • QUOTE: probability density: Any single value of a continuous valued variable usually has zero probability and only a finite range of values has a nonzero probability. Probability of a continuous variable can be characterised by probability density which is defined to be the probability of a range divided by the size of the range.
  • QUOTE: volume: In analogy to physical mass, density and volume, the size of range of continuous valued variables can be called volume. See probability density.

1986

• (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
  • 'Definition 3.2.1. A real-valued function whose domain is the sample space S is called a random variable. We denote random variables by uppercase letters, often [math]\displaystyle{ X }[/math], Y, or Z.

    If the range of the mapping contains either a finite or countably infinite number of values, the random variable is said to be discrete ; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be continuous.

    Associated with each discrete random variable [math]\displaystyle{ Y }[/math] is a probability density function (or pdf). “f_Y(y). By definition, f_Y(y) is the sum of all the probabilities associated with outcomes in [math]\displaystyle{ S }[/math] that get mapped into [math]\displaystyle{ y }[/math] by the random variable Y. That is.

    • f_Y(y) = P({s(∈)S |Y(s) = y})
  • Conceptually, f_Y(y) describes the probability structure induced on the real line by the random variable Y.

    For notational simplicity, we will delete all references to [math]\displaystyle{ s }[/math] and [math]\displaystyle{ S }[/math] and write: f_Y(y) = P(Y(s)=y). In other words, f_Y(y) is the "probability that the random variable Y takes on the value y."

    Associated with each continuous random variable [math]\displaystyle{ Y }[/math] is also a probability density function, f_Y(y), but f_Y(y) in this case is not the probability that the random variable [math]\displaystyle{ Y }[/math] takes on the value y. Rather, f_Y(y) is a continuous curve having the property that for all [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math],

    • P(a ≤ [math]\displaystyle{ Y }[/math] ≤ b) = P({s(∈)S| [math]\displaystyle{ a }[/math] ≤ Y(s) ≤ b}) = Integral(a,b). “f_Y(y) dy]

1972

• (Ord, 1972) ⇒ J. K. Ord. (1972). “Families of Frequency Distributions.” Griffin. ISBN:0-85264-137-0

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