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
27 Related Articles

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.

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

Dual of BCH is an independent source

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