Editing Typical set (section)

===Properties===
An essential characteristic of the typical set is that, if one draws a large number ''n'' of independent random samples from the distribution ''X'', the resulting sequence (''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>) is very likely to be a member of the typical set, even though the typical set comprises only a small fraction of all the possible sequences. Formally, given any <math>\varepsilon>0</math>, one can choose ''n'' such that:
#The probability of a sequence from ''X''<sup>(n)</sup> being drawn from ''A''<sub>''ε''</sub><sup>(''n'')</sup> is greater than 1&nbsp;&minus;&nbsp;''ε'', i.e. <math>Pr[x^{(n)} \in A_\epsilon^{(n)}] \geq 1 - \varepsilon </math>
#<math>\left| {A_\varepsilon}^{(n)} \right| \leqslant 2^{n(H(X)+\varepsilon)}</math>
#<math>\left| {A_\varepsilon}^{(n)} \right| \geqslant (1-\varepsilon)2^{n(H(X)-\varepsilon)}</math>
#If the distribution over <math>\mathcal{X}</math> is not uniform, then the fraction of sequences that are typical is
::<math>\frac{|A_\epsilon^{(n)}|}{|\mathcal{X}^{(n)}|} \equiv \frac{2^{nH(X)}}{2^{n\log_2|\mathcal{X}|}} = 2^{-n(\log_2|\mathcal{X}|-H(X))} \rightarrow 0  </math>

::as ''n'' becomes very large, since <math>H(X) < \log_2|\mathcal{X}|,</math> where <math>|\mathcal{X}|</math> is the [[cardinality]] of <math>\mathcal{X}</math>.

For a general stochastic process {''X''(''t'')} with AEP, the (weakly) typical set can be defined similarly with ''p''(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>) replaced by ''p''(''x''<sub>0</sub><sup>''τ''</sup>) (i.e. the probability of the sample limited to the time interval [0,&nbsp;''τ'']), ''n'' being the [[degrees of freedom (physics and chemistry)|degree of freedom]] of the process in the time interval and ''H''(''X'') being the [[entropy rate]]. If the process is continuous valued, [[differential entropy]] is used instead.