binomial distribution

The Bernoulli distribution models the outcome of a single experiment that can result in only two mutually exclusive events: success or failure. The Bernoulli distribution is defined by the probability mass function:

\[
b(x;p) = p^{x}\\,(1-p)^{\\,1-x}
\]

The binomial distribution models the probability of obtaining a given number of successes in independent Bernoulli trials with constant probability p. Formally, the binomial distribution is expressed as

\[
b(x; n, p) = \binom{n}{x} p^{x} q^{n – x}
\]

The Poisson distribution is a discrete probability distribution that describes how many times a specific event may occur within a fixed period of time or space. Formally, it is expressed as:

\[
p(x; \lambda) = P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}
\]