In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. [1] This property is usually abbreviated as i.i.d., iid, or IID. IID was first defined in statistics and finds application in different fields such as data mining and signal processing.
Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of independent, identically distributed (IID) random data points."
In other words, the terms random sample and IID are synonymous. In statistics, "random sample" is the typical terminology, but in probability, it is more common to say "IID."
Independent and identically distributed random variables are often used as an assumption, which tends to simplify the underlying mathematics. In practical applications of statistical modeling, however, this assumption may or may not be realistic. [3]
The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4]
The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically distributed" implies that an element in the sequence is independent of the random variables that came before it. In this way, an i.i.d. sequence is different from a Markov sequence, where the probability distribution for the nth random variable is a function of the previous random variable in the sequence (for a first-order Markov sequence). An i.i.d. sequence does not imply the probabilities for all elements of the sample space or event space must be the same. [5] For example, repeated throws of loaded dice will produce a sequence that is i.i.d., despite the outcomes being biased.
In signal processing and image processing, the notion of transformation to i.i.d. implies two specifications, the "i.d." part and the "i." part:
i.d. – The signal level must be balanced on the time axis.
i. – The signal spectrum must be flattened, i.e. transformed by filtering (such as deconvolution) to a white noise signal (i.e. a signal where all frequencies are equally present).
Suppose that the random variables and are defined to assume values in . Let and be the cumulative distribution functions of and , respectively, and denote their joint cumulative distribution function by .
Two random variables and are identically distributed if and only if [6] .
Two random variables and are independent if and only if . (See further Independence (probability theory) § Two random variables.)
Two random variables and are i.i.d. if they are independent and identically distributed, i.e. if and only if
(Eq.1) |
The definition extends naturally to more than two random variables. We say that random variables are i.i.d. if they are independent (see further Independence (probability theory) § More than two random variables) and identically distributed, i.e. if and only if
(Eq.2) |
where denotes the joint cumulative distribution function of .
In probability theory, two events, and , are called independent if and only if . In the following, is short for .
Suppose there are two events of the experiment, and . If , there is a possibility . Generally, the occurrence of has an effect on the probability of — this is called conditional probability. Additionally, only when the occurrence of has no effect on the occurrence of , there is .
Note: If and , then and are mutually independent which cannot be established with mutually incompatible at the same time; that is, independence must be compatible and mutual exclusion must be related.
Suppose , , and are three events. If , , , and are satisfied, then the events , , and are mutually independent.
A more general definition is there are events, . If the probabilities of the product events for any events are equal to the product of the probabilities of each event, then the events are independent of each other.
A sequence of outcomes of spins of a fair or unfair roulette wheel is i.i.d. One implication of this is that if the roulette ball lands on "red", for example, 20 times in a row, the next spin is no more or less likely to be "black" than on any other spin (see the gambler's fallacy).
Toss a coin 10 times and record how many times the coin lands on heads.
Such a sequence of two possible i.i.d. outcomes is also called a Bernoulli process.
Roll a die 10 times and record how many times the result is 1.
Choose a card from a standard deck of cards containing 52 cards, then place the card back in the deck. Repeat this 52 times. Record the number of kings that appear.
Many results that were first proven under the assumption that the random variables are i.i.d. have been shown to be true even under a weaker distributional assumption.
The most general notion which shares the main properties of i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti.[ citation needed ] Exchangeability means that while variables may not be independent, future ones behave like past ones — formally, any value of a finite sequence is as likely as any permutation of those values — the joint probability distribution is invariant under the symmetric group.
This provides a useful generalization — for example, sampling without replacement is not independent, but is exchangeable.
In stochastic calculus, i.i.d. variables are thought of as a discrete time Lévy process: each variable gives how much one changes from one time to another. For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process.
One may generalize this to include continuous time Lévy processes, and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the Wiener process is the limit of the Bernoulli process.
Machine learning utilizes the vast amounts of data currently available to deliver faster and more accurate results. [7] To train machine learning models effectively, it is crucial to use historical data that is broadly generalizable. If the training data is not representative of the overall situation, the model's performance on new, unseen data may be inaccurate.
The i.i.d., or independent and identically distributed hypothesis, allows for a significant reduction in the number of individual cases required in the training sample.
This assumption simplifies mathematical maximization calculations. In optimization problems, the assumption of independent and identical distribution simplifies the calculation of the likelihood function. Due to the independence assumption, the likelihood function can be expressed as:
To maximize the probability of the observed event, the log function is applied to maximize the parameter θ. Specifically, it computes:
where
Computers are very efficient at performing multiple additions, but not as efficient at performing multiplications. This simplification enhances computational efficiency. The log transformation, in the process of maximizing, converts many exponential functions into linear functions.
There are two main reasons why this hypothesis is practically useful with the central limit theorem:
A random variable is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which
The likelihood function is the joint probability mass of observed data, but viewed as a function of the parameters of a statistical model. That is, the likelihood function , which gives the likelihood of a vector of parameters under the assumption that a set of observed data is true, is numerically the same as the probability function , which gives the probability of a set of data under the assumption that a vector of parameters is true.
In probability theory and statistics, the chi-squared distribution with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables.
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.
In probability theory, the law of large numbers (LLN) is a mathematical theorem that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean.
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
In probability theory and statistics, a Gaussian process is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. Sometimes loosely referred to as "the" exponential family, this class of distributions is distinct because they all possess a variety of desirable properties, most importantly the existence of a sufficient statistic.
In probability theory and statistics, the cumulantsκn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
In numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in with a density.
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for sampling from a specified multivariate probability distribution when direct sampling from the joint distribution is difficult, but sampling from the conditional distribution is more practical. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
The Kruskal–Wallis test by ranks, Kruskal–Wallis test, or one-way ANOVA on ranks is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann–Whitney U test, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA).
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
In statistics, an exchangeable sequence of random variables is a sequence X1, X2, X3, ... whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences
In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF).
In probability and statistics, the class of exponential dispersion models (EDM), also called exponential dispersion family (EDF), is a set of probability distributions that represents a generalisation of the natural exponential family. Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference.
Although the term well-behaved statistic often seems to be used in the scientific literature in somewhat the same way as is well-behaved in mathematics it can also be assigned precise mathematical meaning, and in more than one way. In the former case, the meaning of this term will vary from context to context. In the latter case, the mathematical conditions can be used to derive classes of combinations of distributions with statistics which are well-behaved in each sense.
Bayesian hierarchical modelling is a statistical model written in multiple levels that estimates the parameters of the posterior distribution using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. The result of this integration is the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired.