Editing Typical set (section)

In [[information theory]], the '''typical set''' is a set of sequences whose [[probability]] is close to two raised to the negative power of the [[Information entropy|entropy]] of their source distribution. That this set has total [[probability]] close to one is a consequence of the [[asymptotic equipartition property]] (AEP) which is a kind of [[law of large numbers]]. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.

This has great use in [[data compression|compression]] theory as it provides a theoretical means for compressing data, allowing us to represent any sequence ''X''<sup>''n''</sup> using ''nH''(''X'') bits on average, and, hence, justifying the use of entropy as a measure of information from a source.

The AEP can also be proven for a large class of [[stationary ergodic process]]es, allowing typical set to be defined in more general cases.

Additionally, the typical set concept is foundational in understanding the limits of data transmission and error correction in communication systems. By leveraging the properties of typical sequences, efficient coding schemes like Shannon's [[Shannon's source coding theorem|source coding theorem]] and [[Noisy-channel coding theorem|channel coding theorem]] are developed, enabling near-optimal data compression and reliable transmission over noisy channels.