Probability distribution
Mathematical function for the probability a given outcome occurs in an experiment / From Wikipedia, the free encyclopedia
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In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.[1][2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).[3]
For instance, if 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 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.
Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.
A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often represented in notation by is the set of all possible outcomes of a random phenomenon being observed; it may be any set: A set of real numbers, a set of descriptive labels, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip would be Ω = { "heads", "tails" } .
To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six digits “1” to “6”, corresponding to the number of dots on the die, has the probability The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is
In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale has many digits of precision. The probability that it weighs exactly 500 g is zero, as it will most likely have some non-zero decimal digits. Nevertheless, one might demand, in quality control, that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability, and this demand is less sensitive to the accuracy of measurement instruments.
Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval.[4] An alternative description of the distribution is by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., for some ). The cumulative distribution function is the area under the probability density function from to as described by the picture to the right.[5]
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function whose input space is a σ-algebra, and gives a real number probability as its output, particularly, a number in .
The probability function can take as argument subsets of the sample space itself, as in the coin toss example, where the function was defined so that P(heads) = 0.5 and P(tails) = 0.5. However, because of the widespread use of random variables, which transform the sample space into a set of numbers (e.g., , ), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets),[6] and all probability distributions discussed in this article are of this type. It is common to denote as the probability that a certain value of the variable belongs to a certain event .[7][8]
The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is:
- , so the probability is non-negative
- , so no probability exceeds
- for any countable disjoint family of sets
The concept of probability function is made more rigorous by defining it as the element of a probability space , where is the set of possible outcomes, is the set of all subsets whose probability can be measured, and is the probability function, or probability measure, that assigns a probability to each of these measurable subsets .[9]
Probability distributions usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function. On the other hand, absolutely continuous probability distributions are applicable to 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. In the absolutely continuous case, probabilities are described by a probability density function, and the probability distribution is by definition the integral of the probability density function.[7][4][8] The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.
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. A univariate distribution gives the probabilities of a single random variable taking on various different values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. A commonly encountered multivariate distribution is the multivariate normal distribution.
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.[10]
Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.[1]
Basic terms
- Random variable: takes values from a sample space; probabilities describe which values and set of values are taken more likely.
- Event: set of possible values (outcomes) of a random variable that occurs with a certain probability.
- Probability function or probability measure: describes the probability that the event occurs.[11]
- Cumulative distribution function: function evaluating the probability that will take a value less than or equal to for a random variable (only for real-valued random variables).
- Quantile function: the inverse of the cumulative distribution function. Gives such that, with probability , will not exceed .
Discrete probability distributions
- Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
- Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.
- Frequency distribution: a table that displays the frequency of various outcomes in a sample.
- Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size).
- Categorical distribution: for discrete random variables with a finite set of values.
Absolutely continuous probability distributions
- Absolutely continuous probability distribution: for many random variables with uncountably many values.
- Probability density function (pdf) or probability density: 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.
Related terms
- Support: set of values that can be assumed with non-zero probability (or probability density in the case of a continuous distribution) by the random variable. For a random variable , it is sometimes denoted as .
- Tail:[12] the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form , or a union thereof.
- Head:[12] the region where the pmf or pdf is relatively high. Usually has the form .
- Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
- Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
- Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
- Quantile: the q-quantile is the value such that .
- Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
- Standard deviation: the square root of the variance, and hence another measure of dispersion.
- Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
- Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
- Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable with regard to a probability distribution is defined as
The cumulative distribution function of any real-valued random variable has the properties:
- is non-decreasing;
- is right-continuous;
- ;
- and ; and
- .
Conversely, any function that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.[13]
Any probability distribution can be decomposed as the mixture of a discrete, an absolutely continuous and a singular continuous distribution,[14] and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions.
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values[15] (almost surely)[16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum:
where is a countable set with . Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function . In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if for , the sum of probabilities would be .
Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution.[3] When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.
Cumulative distribution function
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form
The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.
Dirac delta representation
A discrete probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For any outcome , let be the Dirac measure concentrated at . Given a discrete probability distribution, there is a countable set with and a probability mass function . If is any event, then
or in short,
Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function , where
which means
for any event [17]
Indicator-function representation
For a discrete random variable , let be the values it can take with non-zero probability. Denote
These are disjoint sets, and for such sets
It follows that the probability that takes any value except for is zero, and thus one can write as
except on a set of probability zero, where is the indicator function of . This may serve as an alternative definition of discrete random variables.
One-point distribution
A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution. Expressed formally, the random variable has a one-point distribution if it has a possible outcome such that [18] All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.