Distribution ensemble

In <a href="/facts/Cryptography/e94PEVau">cryptography</a>, a distribution ensemble or probability ensemble is a family of distributions or <a href="/facts/Random_variable/TwTBXnLT">random variables</a> 
 
 
 
 X
 =
 {
 
 X
 
 i
 
 
 
 }
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle X=\{X_{i}\}_{i\in I}}
 
 where 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is a (<a href="/facts/Countable/Wj4DmWPv">countable</a>) index set, and each 
 
 
 
 
 X
 
 i
 
 
 
 
 {\displaystyle X_{i}}
 
 is a random variable, or <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a>. Often 
 
 
 
 I
 =
 
 N
 
 
 
 {\displaystyle I=\mathbb {N} }
 
 and it is required that each 
 
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle X_{n}}
 
 have a certain property for n sufficiently large.
For example, a uniform ensemble 
 
 
 
 U
 =
 {
 
 U
 
 n
 
 
 
 }
 
 n
 ∈
 
 N
 
 
 
 
 
 {\displaystyle U=\{U_{n}\}_{n\in \mathbb {N} }}
 
 is a distribution ensemble where each 
 
 
 
 
 U
 
 n
 
 
 
 
 {\displaystyle U_{n}}
 
 is <a href="/facts/Uniform_distribution_(discrete)/M52SQP5n">uniformly distributed</a> over strings of length n. In fact, many applications of probability ensembles implicitly assume that the probability spaces for the random variables all coincide in this way, so every probability ensemble is also a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a>.

Distribution ensemble open-in-new

Distribution ensemble