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In probability theory and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a collection of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s is independent and identically distributed if each random variable has the same
probability distribution 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. It is a mathematical description of a random phenomenon i ...
as the others and all are mutually independent. 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.


Introduction

In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the sample are taken from the same probability distribution. * Independent means that the sample items are all independent events. In other words, they aren’t connected to each other in any way. In other words, knowledge of the value of one variable gives no information about the value of the other and vice versa.


Application

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, the assumption may or may not be realistic. The i.i.d. assumption is also used in 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. Often the i.i.d. assumption 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 A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
, where the probability distribution for the ''n''th 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. For example, repeated throws of loaded dice will produce a sequence that is i.i.d., despite the outcomes being biased.


Definition


Definition for two random variables

Suppose that the random variables X and Y are defined to assume values in I \subseteq \mathbb. Let F_X(x) = \operatorname(X\leq x) and F_Y(y) = \operatorname(Y\leq y) be the cumulative distribution functions of X and Y, respectively, and denote their joint cumulative distribution function by F_(x,y) = \operatorname(X\leq x \land Y\leq y). Two random variables X and Y are identically distributed if and only if F_X(x)=F_Y(x) \, \forall x \in I. Two random variables X and Y are independent if and only if F_(x,y) = F_(x) \cdot F_(y) \, \forall x,y \in I. (See further .) Two random variables X and Y are i.i.d. if they are independent ''and'' identically distributed, i.e. if and only if


Definition for more than two random variables

The definition extends naturally to more than two random variables. We say that n random variables X_1,\ldots,X_n are i.i.d. if they are independent (see further ) ''and'' identically distributed, i.e. if and only if where F_(x_1,\ldots,x_n) = \operatorname(X_1\leq x_1 \land \ldots \land X_n\leq x_n) denotes the joint cumulative distribution function of X_1,\ldots,X_n.


Definition for independence

In probability theory, two events A, B are called independent if and only if P(A and B) = P(A)P(B). In the following P(AB) is short for P(A and B). Suppose there are the two events of the experiment A, B if P(A) > 0, there is possibility P(B, A). Generally, the occurrence of A has an effect on the probability of B, which is called conditional probability, and only when the occurrence of A has no effect on the occurrence of B, there is P(B, A) = P(B). Note: If P(A) > 0, P(B) > 0 then A, B 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 A, B, C are three events. If P(AB) = P(A)P(B), P(BC) = P(B)P(C), P(AC) = P(A)P(C), P(ABC) = P(A)P(B)P(C) are satisfied, then the events A, B, C are independent of each other. A more general definition is there are n events, A1, A2,...,An. If the probabilities of the product events for any 2, 3, ..., n events are equal to the product of the probabilities of each event, then the events A1, A2, ..., An are independent of each other.


Examples


Example 1

A sequence of outcomes of spins of a fair or unfair
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
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). A sequence of fair or loaded dice rolls is i.i.d. A sequence of fair or unfair coin flips is i.i.d. In signal processing and
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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).


Example 2

Toss a coin 10 times and record how many times does the coin lands on head. # Independent – each outcome of landing will not affect the other outcome, which means the 10 results are independent from each other. # Identically Distributed – if the coin is a homogeneous material, each time the probability for head is 0.5, which means the probability is identical for each time.


Example 3

Roll a dice 10 times and record how many time the result is 1. # Independent – each outcome of the dice will not affect the next one, which means the 10 results are independent from each other. # Identically Distributed – if the dice is a homogeneous material, each time the probability for the number 1 is 1/6, which means the probability is identical for each time.


Example 4

Choose a card from a standard deck of cards containing 52 cards, then place the card back in the deck. Repeat it for 52 times. Record the number of King appears # Independent – each outcome of the card will not affect the next one, which means the 52 results are independent from each other. # Identically Distributed – after drawing one card from it, each time the probability for King is 4/52, which means the probability is identical for each time.


Generalizations

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.


Exchangeable random variables

The most general notion which shares the main properties of i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti. 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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of those values – the
joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
is invariant under the symmetric group. This provides a useful generalization – for example, sampling without replacement is not independent, but is exchangeable.


Lévy process

In stochastic calculus, i.i.d. variables are thought of as a
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
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 In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. Th ...
. 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.


In machine learning


Why assume the data in machine learning are independent and identically distributed?

Machine learning uses currently acquired massive quantities of data to deliver faster, more accurate results. Therefore, we need to use historical data with overall representativeness. If the data obtained is not representative of the overall situation, then the rules will be summarized badly or wrongly. Through i.i.d. hypothesis, the number of individual cases in the training sample can be greatly reduced. This assumption makes maximization very easy to calculate mathematically. Observing the assumption of independent and identical distribution in mathematics simplifies the calculation of the likelihood function in optimization problems. Because of the assumption of independence, the likelihood function can be written like this :l(\theta) = P(x_1, x_2, x_3,...,x_n, \theta) = P(x_1, \theta) P(x_2, \theta) P(x_3, \theta) ... P(x_n, \theta) In order to maximize the probability of the observed event, take the log function and maximize the parameter θ. That is to say, to compute: :\mathop\limits_\theta \log(l(\theta)) where :\log(l(\theta)) = \log(P(x_1, \theta)) + \log(P(x_2, \theta)) + \log(P(x_3, \theta)) + ... + \log(P(x_n, \theta)) The computer is very efficient to calculate multiple additions, but it is not efficient to calculate the multiplication. This simplification is the core reason for the increase in computational efficiency. And this Log transformation is also in the process of maximizing, turning many exponential functions into linear functions. For two reasons, this hypothesis is easy to use the central limit theorem in practical applications. # Even if the sample comes from a more complex non-Gaussian distribution, it can also approximate well. Because it can be simplified from the central limit theorem to Gaussian distribution. For a large number of observable samples, "the sum of many random variables will have an approximately normal distribution". # The second reason is that the accuracy of the model depends on the simplicity and representative power of the model unit, as well as the data quality. Because the simplicity of the unit makes it easy to interpret and scale, and the representative power + scale out of the unit improves the model accuracy. Like in a deep neural network, each neuron is very simple but has strong representative power, layer by layer to represent more complex features to improve model accuracy.


See also

* De Finetti's theorem * Pairwise independent variables * Central limit theorem


References


Further reading

* {{DEFAULTSORT:Independent And Identically-Distributed Random Variables Independence (probability theory) Statistical theory