# Probability mass function

mass functionprobability massmassdiscrete probability functiondiscrete probability spacemass functionspmfprobability mass distributionprobability mass functions
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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### Mode (statistics)

modemodalmodes
The value of the random variable having the largest probability mass is called the mode.
If X is a discrete random variable, the mode is the value x (i.e, X = x) at which the probability mass function takes its maximum value.

### 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 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.

### 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.

### 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. This statement isn't true for a continuous random variable X, for which for any possible x: in fact, by definition, a continuous random variable can have an infinite set of possible values and thus the probability it has a single particular value x is equal to.
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
The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable 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.

### Binomial distribution

binomialbinomial modelBinomial probability
There are three major distributions associated, the Bernoulli distribution, the Binomial distribution and the geometric distribution.
The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:

### Joint probability distribution

joint distributionjoint probabilitymultivariate 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).

### Bernoulli distribution

BernoulliBernoulli random variableBernoulli random variables
There are three major distributions associated, the Bernoulli distribution, the Binomial distribution and the geometric distribution.
The probability mass function f of this distribution, over possible outcomes k, is

### Categorical distribution

categoricalcategorical probability distributioncategorical variable
Other distributions that can be modeled using a probability mass function is the Categorical distribution (also known as the generalized Bernoulli distribution)and the multinomial distribution.
Both forms have very similar-looking probability mass functions (PMFs), which both make reference to multinomial-style counts of nodes in a category.

### Multinomial distribution

multinomialmultinomially distributed
Other distributions that can be modeled using a probability mass function is the Categorical distribution (also known as the generalized Bernoulli distribution)and 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)

variablevariablesscalar
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 PDF must be integrated over an interval to yield a probability.

### 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

measuremeasure spacesprobability space
Now suppose that is a measure space equipped with the counting measure μ.

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.

### 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.

### Infinite set

infiniteinfinitelyinfinitely many
This statement isn't true for a continuous random variable X, for which for any possible x: in fact, by definition, a continuous random variable can have an infinite set of possible values and thus the probability it has a single particular value x is equal to.