# 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.