probability mass function

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 hypergeometric distribution is a discrete probability distribution that describes the number of successes drawn from a finite population without replacement. Formally, it is defined as:

\[
P(X = x) = \frac{\binom{K}{x}\\,\binom{N-K}{n-x}}{\binom{N}{n}}
\]

The geometric distribution describes the number of independent trials required to observe the first success in a repeated experiment. It is defined by the probability mass function:

\[
P(X = k) = (1 – p)^{\\,k – 1}\\, p, \qquad k = 1, 2, 3, \dots
\]