# Pairwise independence

**pairwise independentindependent sourcepairwiseℓ''-wise independent**

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.wikipedia

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### Independence (probability theory)

**independentstatistically independentindependence**

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.

A finite set of n random variables is pairwise independent if and only if every pair of random variables is independent.

### MAXEkSAT

**MAX-EkSAT**

k-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem MAXEkSAT.

is an ℓ-wise independent source if, for a uniformly chosen random (x 1, x 2, ..., x n ) ∈ S, x 1, x 2, ..., x n are ℓ-wise independent random variables.

### Pairwise

Pairwise

Pairwise independence of random variables

### Message authentication code

**MACmessage authentication codesauthenticate**

k-wise independence is used in the proof that k-independent hashing functions are secure unforgeable message authentication codes.

Universal hashing and in particular pairwise independent hash functions provide a secure message authentication code as long as the key is used at most once.

### Probability theory

**theory of probabilityprobabilityprobability theorist**

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.

### Random variable

**random variablesrandom variationrandom**

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.

### Variance

**sample variancepopulation variancevariability**

Pairwise independent random variables with finite variance are uncorrelated.

### Uncorrelatedness (probability theory)

**uncorrelated**

Pairwise independent random variables with finite variance are uncorrelated.

### Joint probability distribution

**joint distributionjoint probabilitymultivariate distribution**

A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) satisfies

### Modular arithmetic

**modulomodcongruent**

However, X, Y, and Z are not mutually independent, since the left side equalling for example 1/4 for (x, y, z) = (0, 0, 0) while the right side equals 1/8 for (x, y, z) = (0, 0, 0). In fact, any of \{X,Y,Z\} is completely determined by the other two (any of X, Y, Z is the sum (modulo 2) of the others).

### K-independent hashing

**-independence2-universal3-wise independent**

k-wise independence is used in the proof that k-independent hashing functions are secure unforgeable message authentication codes.

### Disjoint sets

**disjointpairwise disjointdisjoint set**

Pairwise disjoint

### Universal hashing

**universal hash functionuniversaluniversal hash**

An even stronger condition is pairwise independence: we have this property when we have the probability that x,y will hash to any pair of hash values z_1, z_2 is as if they were perfectly random:.

### Rolling hash

**rolling checksumBuzhashRabin-Karp rolling hash**

At best, rolling hash values are pairwise independent or strongly universal.

### Stochastic dynamic programming

**backward recursion**

A gambler has $2, she is allowed to play a game of chance 4 times and her goal is to maximize her probability of ending up with a least $6. If the gambler bets $b on a play of the game, then with probability 0.4 she wins the game, recoup the initial bet, and she increases her capital position by $b; with probability 0.6, she loses the bet amount $b; all plays are pairwise independent.

### List of mathematical examples

**Mathematical examples**

Pairwise independence of random variables need not imply mutual independence.

### Lévy process

**increments are stationary and independentLévyLévy measure**

To call the increments of a process independent means that increments X s − X t and X u − X v are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent.

### Law of large numbers

**strong law of large numbersweak law of large numbersLaws of large numbers**

Mutual independence of the random variables can be replaced by pairwise independence in both versions of the law.

### Central limit theorem

**limit theoremsA proof of the central limit theoremcentral limit**

are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent.

### Multivariate normal distribution

**multivariate normalbivariate normal distributionjointly normally distributed**

This implies that any two or more of its components that are pairwise independent are independent.

### Randomized algorithm

**probabilisticprobabilistic algorithmderandomization**

the exploitation of limited independence in the random variables used by the algorithm, such as the pairwise independence used in universal hashing

### List of statistics articles

**list of statistical topicslist of statistics topics**

Pairwise independence

### Dual of BCH is an independent source

treated as linear operators, their dual operators turns their input into an \ell-wise independent source.