Probability mass function

mass functionprobability massmassprobability mass functionsdiscrete probability functiondiscrete probability spacemass functionsprobability mass distribution
In probability and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.wikipedia
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Probability density function

probability densitydensity functiondensity
A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the probability density function are not probabilities as such: a pdf must be integrated over an interval to yield a probability. the distribution of X and the probability density function of X with respect to the counting measure.
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 (PMF) rather than the density.

Mode (statistics)

modemodalmodes
The value of the random variable having the largest probability mass is called the mode.
It is the value x at which its probability mass function takes its maximum value.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. the distribution of X and the probability density function of X with respect to the counting measure.
A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
Since the image of X is countable, the probability mass function f_X(x) is zero for all but a countable number of values of x. The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable, when it is meaningful because there is a natural ordering, is also discontinuous.
It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions.

Probability space

probability measuresGaussian measureoutcomes
Suppose that is a probability space
Probabilities can be ascribed to points of \Omega by the probability mass function such that.

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
*An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution.
The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables).

Categorical distribution

categoricalcategorical probability distributioncategorical variable
*If the discrete distribution has two or more categories one of which may occur, whether or not these categories have a natural ordering, when there is only a single trial (draw) this is a categorical distribution.
Both forms have very similar-looking probability mass functions (PMF's), which both make reference to multinomial-style counts of nodes in a category.

Multinomial distribution

multinomialmultinomially distributed
*An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution.
The probability mass function of this multinomial distribution is:

Probability theory

theory of probabilityprobabilityprobability theorist
In probability and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.

Statistics

statisticalstatistical analysisstatistician
In probability and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.

Variable (computer science)

variablevariablesextent
The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

Multivariate random variable

random vectorvectormultivariate
The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. When there is a natural order among the potential outcomes x, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of X. That is, f_X may be defined for all real numbers and f_X(x)=0 for all as shown in the figure.

Domain of a function

domaindomainsdomain of definition
The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

Integral

integrationintegral calculusdefinite integral
A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the probability density function are not probabilities as such: a pdf must be integrated over an interval to yield a probability.

Sample space

event spacespacesample spaces
Suppose that for A is a discrete random variable defined on a sample space S.

Image (mathematics)

imagepreimageinverse image
When there is a natural order among the potential outcomes x, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of X. That is, f_X may be defined for all real numbers and f_X(x)=0 for all as shown in the figure.

Real number

realrealsreal-valued
When there is a natural order among the potential outcomes x, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of X. That is, f_X may be defined for all real numbers and f_X(x)=0 for all as shown in the figure.

Counting measure

the distribution of X and the probability density function of X with respect to the counting measure.

Sigma-algebra

σ-algebraσ-algebrasigma algebra
and that is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of B. In this setting, a random variable is discrete provided its image is countable.

Pushforward measure

push forwardpush-forward measurepushforward
The pushforward measure X_{*}(P)—called a distribution of X in this context—is a probability measure on B whose restriction to singleton sets induces a probability mass function since for each b \in B.

Measure space

measureprobability spacemeasure spaces.
Now suppose that is a measure space equipped with the counting measure μ.

Radon–Nikodym theorem

Radon–Nikodym derivativedensityRadon-Nikodym derivative
The probability density function f of X with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of X (with respect to the counting measure), so and f is a function from B to the non-negative reals.

Entropy (information theory)

entropyinformation entropyShannon entropy
The measure of information entropy associated with each possible data value is the negative logarithm of the probability mass function for the value:

Poisson distribution

PoissonPoisson-distributedPoissonian
This equation is the probability mass function (PMF) for a Poisson distribution.