Definitions
Nearby Words

# Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable (when the variable is discrete), or the probability of the value falling within a particular interval (when the variable is continuous). The probability function describes the range of possible values that a random variable can attain and the probability that the value of the random variable is within any (measurable) subset of that range.

When the random variable takes values in the set of real numbers, the probability distribution is completely described by the cumulative distribution function, whose value at each real x is the probability that the random variable is smaller than or equal to x.

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

There are various probability distributions that show up in various different applications. One of the more important ones is the normal distribution, which is also known as the Gaussian distribution or the bell curve and approximates many different naturally occuring distributions. The toss of a fair coin yields another familiar distribution, where the possible values are heads or tails, each with probability 1/2.

## Rigorous definitions

In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. That is, probability distributions are probability measures defined over a state space instead of the sample space. A random variable then defines a probability measure on the sample space by assigning a subset of the sample space the probability of its inverse image in the state space. In other words the probability distribution of a random variable is the push forward measure of the probability distribution on the state space.

In other words, given a random variable $X: Omega rightarrow Y$ between a probability space $\left(Omega, mathcal\left\{F\right\}, P\right)$, the sample space, and a measurable space $\left(Y, Sigma\right)$, called the state space, a probability distribution on (Y, Σ) is a probability measure $X_\left\{*\right\}P: Sigma rightarrow \left[0,1\right]$ on the state space where $X_\left\{*\right\}P$ is the push forward measure of P.

### Probability distributions of real-valued random variables

Because a probability distribution Pr on the real line is determined by the probability of being in a half-open interval Pr(ab], the probability distribution of a real-valued random variable X is completely characterized by its cumulative distribution function:

$F\left(x\right) = Pr left\left[X le x right\right] qquad forall x in mathbb\left\{R\right\}.$

#### Discrete probability distribution

A probability distribution is called discrete if its cumulative distribution function only increases in jumps. More precisely, a probability distribution is discrete if there is a finite or countable set whose probability is 1.

For many familiar discrete distributions, the set of possible values is topologically discrete in the sense that all its points are isolated points. But, there are discrete distributions for which this countable set is dense on the real line.

Discrete distributions are characterized by a probability mass function, $p$ such that


Pr left[X = x right] = p(x).

#### Continuous probability distribution

By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous, which means that it belongs to a random variable X for which Pr[X = x ] = 0 for all x in R.

Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function $f$ defined on the real numbers such that


F(x) = Pr left[X le x right] = int_{-infty}^x f(t),dt

Discrete distributions and some continuous distributions (like the devil's staircase) do not admit such a density.

### Terminology

The support of a distribution is the smallest closed interval/set whose complement has probability zero.

The probability density function of the sum of two independent random variables is the convolution of each of their density functions.

The probability density function of the difference of two random variables is the cross-correlation of each of their density functions.

A discrete random variable is a random variable whose probability distribution is discrete. Similarly, a continuous random variable is a random variable whose probability distribution is continuous.

## List of important probability distributions

Certain random variables occur very often in probability theory, in some cases due to their application to many natural and physical processes, and in some cases due to theoretical reasons such as the central limit theorem, the Poisson limit theorem, or properties such as memorylessness or other characterizations. Their distributions therefore have gained special importance in probability theory.

### Discrete distributions

#### With finite support

• The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p.
• The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
• The binomial distribution describes the number of successes in a series of independent Yes/No experiments.
• The degenerate distribution at x0, where X is certain to take the value x0. This does not look random, but it satisfies the definition of random variable. It is useful because it puts deterministic variables and random variables in the same formalism.
• The discrete uniform distribution, where all elements of a finite set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette or a well-shuffled deck of playing cards. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, pseudo-random number generators are used to produce a statistically random discrete uniform distribution.
• The hypergeometric distribution, which describes the number of successes in the first m of a series of n Yes/No experiments, if the total number of successes is known.
• Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
• The Zipf-Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.

### Joint distributions

For any set of independent random variables the probability density function of their joint distribution is the product of their individual density functions.