 In other words, if "X" and "Y" are the random variables describing the probability distributions of the input and the output of the channel, respectively, then the crossovers of the channel are characterized by the conditional probabilities

The capacity formula_1 of the Z-channel formula_2 with the crossover 1 → 0 probability "p", when the input random variable "X" is distributed according to the Bernoulli distribution with probability "formula_3" for the occurrence of 0, is calculated as follows.where formula_7 is the binary entropy function.To find the maximum we differentiate
And we see the maximum is attained for
yielding the following value of formula_1 as a function of "p"

For small "p", the capacity is approximated by

as compared to the capacity formula_13 of the binary symmetric channel with crossover probability "p".