In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. For example, the event of getting a "6" when a die is rolled and the event of getting a "6" the second time are independent. Similarly, two random variables are independent if the conditional probability distribution of either given the observed value of the other is the same as if the other's value had not been observed. Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

Independent events

The standard definition says:

Two events A and B are independent if and only if Pr(A ∩ B) = Pr(A)Pr(B).

Here A ∩ B is the intersection of A and B, that is, it is the event that both events A and B occur. It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

More generally, any collection of events -- possibly more than just two of them -- are mutually independent if and only if for any finite subset A₁, ..., A_n of the collection we have

This is called the multiplication rule for independent events.

If two events A and B are independent, then the conditional probability of A given B is the same as the "unconditional" (or "marginal") probability of A, that is, This article defines some terms which characterize probability distributions of two or more variables. ...

There are at least two reasons why this statement is not taken to be the definition of independence: (1) the two events A and B do not play symmetrical roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved.

When one recalls that the conditional probability Pr(A | B) is given by

(so long as Pr(B) ≠ 0 )

one sees that the statement above is equivalent to

which is the standard definition given above.

Note that independence does not have the same meaning as it does in the vernacular. For example an event is independent of itself if and only if Look up Vernacular in Wiktionary, the free dictionary. ...

that is, if its probability is one or zero. Thus if an event or its complement almost surely occurs, it is independent of itself. For example, if event A is choosing 0.5 from a uniform distribution on the unit interval, A is independent of itself, even though, tautologically, A fully determines A. In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, specifically, in probability theory, the phrase almost surely is a concise, precise way to state except on a set or event of probability measure zero. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ...

Independent random variables

What is defined above is independence of events. In this section we treat independence of random variables. If X is a real-valued random variable and a is a number, then the event that X ≤ a is an event, so it makes sense to speak of its being, or not being, independent of another event. A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, the real numbers may be described informally in several different ways. ...

Two random variables X and Y are independent if and only if for any numbers a and b the events [X ≤ a] (the event of X being less than or equal to a) and [Y ≤ b] are independent events as defined above. Similarly an arbitrary collection of random variables -- possible more than just two of them -- is independent precisely if for any finite collection X₁, ..., X_n and any finite set of numbers a₁, ..., a_n, the events [X₁ ≤ a₁], ..., [X_n ≤ a_n] are independent events as defined above. It has been suggested that this article or section be merged with Logical biconditional. ...

The measure-theoretically inclined may prefer to substitute events [X ∈ A] for events [X ≤ a] in the above definition, where A is any Borel set. That definition is exactly equivalent to the one above when the values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values in any topological space. In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In mathematics, the real numbers may be described informally in several different ways. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

If any two of a collection of random variables are independent, they may nonetheless fail to be mutually independent; this is called pairwise independence. In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. ...

If X and Y are independent, then the expectation operator E has the nice property In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

E[X Y] = E[X] E[Y],

and for the variance we have In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

var(X + Y) = var(X) + var(Y),

so the covariance cov(X,Y) is zero. (The converse of these, i.e. the proposition that if two random variables have a covariance of 0 they must be independent, is not true. See uncorrelated.) In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... In probability theory and statistics, to call two real-valued random variables X and Y uncorrelated means that their correlation is zero, or, equivalently, their covariance is zero. ...

Furthermore, random variables X and Y with distribution functions F_X(x) and F_Y(y), and probability densities f_X(x) and f_Y(y), are independent if and only if the combined random variable (X,Y) has a joint distribution In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

F_X,Y(x,y) = F_X(x)F_Y(y),

or equivalently, a joint density

f_X,Y(x,y) = f_X(x)f_Y(y).

Similar expressions characterise independence more generally for more than two random variables.

Conditionally independent random variables

Main article: Conditional independence In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given...

Intuitively, two random variables X and Y are conditionally independent given Z if, once Z is known, the value of Y does not add any additional information about X. For instance, two measurements X and Y of the same underlying quantity Z are not independent, but they are conditionally independent given Z (unless the errors in the two measurements are somehow connected).

The formal definition of conditional independence is based on the idea of conditional distributions. If X, Y, and Z are discrete random variables, then we define X and Y to be conditionally independent given Z if Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X (written Y | X) is the probability distribution of Y when X is known to be a particular value. ... In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...

P(X ≤x, Y ≤y | Z ≤z) = P(X ≤x | Z ≤ z) · P(Y ≤ y | Z ≤ z)

for all x, y and z such that P(Z ≤ z) > 0. On the other hand, if the random variables are continuous and have a joint probability density function p, then X and Y are conditionally independent given Z if By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

p_XY|Z(x, y | z) = p_X|Z(x | z) · p_Y|Z(y | z)

for all real numbers x, y and z such that p_Z(z) > 0.

If X and Y are conditionally independent given Z, then

P(X = x | Y = y, Z = z) = P(X = x | Z = z)

for any x, y and z with P(Z = z) > 0. That is, the conditional distribution for X given Y and Z is the same as that given Z alone. A similar equation holds for the conditional probability density functions in the continuous case.

Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events.

See also

• Copula (statistics)
• Independent identically-distributed random variables

In statistics, a copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 1]n such that every marginal distribution is uniform on the interval [0, 1]. Sklars theorem is as follows. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

Further reading

• Kirby Faciane (2006). Statistics for Empirical and Quantitative Finance. H.C. Baird: Philadelphia. ISBN 0978820894.