Discrete Memoryless Channel (DMC)

A DMC is a Communication Channel for which

is the same each time the channel is used, it is independent of all past uses

Binary Symmetric Channel (BSC)
Binary Erasure Channel

Lemma

The -th Communication Channel Extension of a DMC with Information Channel Capacity has information capacity

Proof

We take rv input and rv output
Now

Since the channel is memoryless

So,

Thus

Equality is if are independent. This can only be achieved for independent and choosing the probability distribution s.t.