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The probability mass function of binomial distribution is defined in the following way: (18).

The probability mass function for the binomial distribution is defined to be B k ; n, p = n k p k 1 − p n − k, where n k = n ! k ! n − k !, n is the number of trials, k is the number of successes, and p is the probability of success.

$$\end{document} In order to construct a likelihood function for the actual number of cases we assume that the number of observed cases follows a binomial distribution that is defined by a number of N i independent trials where the probability of success is ρ i.

The sample is drawn without replacement, so, strictly speaking, the binomial distribution is not applicable.

The binomial distribution is now a conditional probability, given the value of r.

A standardized binomial distribution is a binomial distribution scaled and translated so that it has zero mean and unit variance.

Thus, the binomial distribution is a generalization of the Bernoulli distribution.

Binomial distribution is a fundamental statistical assumption about sampling process.

Therefore, the binomial distribution is a conservative model for the sampling process.

An extension of the binomial distribution is the beta-binomial distribution.

While the negative binomial distribution can be defined in the same way for k-mers and genomic elements, differences exist in the way the number of "failures" and the success probability are calculated.