# Probability distribution

**distributioncontinuous probability distributiondiscrete probability distributionprobability distributionscontinuousdiscretecontinuous random variablecontinuous distributiondiscrete distributiondistributions**

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.wikipedia

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### Randomness

**randomchancerandomly**

In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events.

Individual random events are by definition unpredictable, but since they often follow a probability distribution, the frequency of different outcomes over numerous events (or "trials") is predictable.

### Probability theory

**theory of probabilityprobabilityprobability theorist**

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

### Statistics

**statisticalstatistical analysisstatistician**

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Numerical descriptors include mean and standard deviation for continuous data types (like income), while frequency and percentage are more useful in terms of describing categorical data (like education).

### Random variable

**random variablesrandom variationrandom**

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for

A random variable has a probability distribution, which specifies the probability of its values.

### Probability density function

**probability densitydensity functiondensity**

On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0).

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

### Probability mass function

**mass functionprobability massmass**

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.

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.

### Univariate distribution

**univariateuni**

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate.

In statistics, a univariate distribution is a probability distribution of only one random variable.

### Joint probability distribution

**joint distributionjoint probabilitymultivariate distribution**

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate.

Given random variables X,Y,\ldots, that are defined on a probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range or discrete set of values specified for that variable.

### Binomial distribution

**binomialbinomial modelBinomial probability**

Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q = 1 − p).

### Categorical distribution

**categoricalcategorical probability distributioncategorical variable**

In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution ) is a discrete probability distribution that describes the possible results of a random variable that can take on one of K possible categories, with the probability of each category separately specified.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval.

In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

### Expected value

**expectationexpectedmean**

In probability theory, the expected value of a random variable is a key aspect of its probability distribution.

### Heavy-tailed distribution

**heavy tailsheavy-tailedheavy tail**

In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution.

### Median

**averagesample medianmedian-unbiased estimator**

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution).

### Statistical dispersion

**dispersionvariabilityspread**

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.

### Symmetric probability distribution

**symmetricsymmetric distributionSymmetry**

In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass function is reflected around a vertical line at some value of the random variable represented by the distribution.

### Skewness

**skewedskewskewed distribution**

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.

### Mode (statistics)

**modemodalmodes**

The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points x 1, x 2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

### Event (probability theory)

**eventeventsrandom event**

In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events.

For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers.

### Standard deviation

**standard deviationssample standard deviationSD**

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.

### Kurtosis

**excess kurtosisleptokurticplatykurtic**

In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable.

### Poisson distribution

**PoissonPoisson-distributedPoissonian**

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

In probability theory and statistics, the Poisson distribution (in English often rendered ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

### Negative binomial distribution

**negative binomialGamma-Poisson distributioninverse binomial distribution**

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.

### Geometric distribution

**geometricgeometrically distributed geometrically distributed**

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:

### Stochastic process

**stochastic processesstochasticrandom process**

More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

all have the same probability distribution.