# 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-algebra&sigma;-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.