expected value

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:

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

A continuous random variable \( X \) is said to follow a uniform distribution on the interval \(A = [a, b]\) if its probability density function is constant on that interval. Formally we have:

\[
U(x;a,b) =
\begin{cases}
\dfrac{1}{\\,b – a\\,} & x \in A \\\\[0.6em]
0 & x \notin A
\end{cases}
\]

The gamma distribution is a continuous probability distribution defined on the positive half-line. Its probability density function is given by

\[
G(x;\alpha,\beta)=
\begin{cases}
\dfrac{1}{\beta^{\alpha}\\,\Gamma(\alpha)}\\, x^{\alpha – 1}\\, e^{-x/\beta} & x>0\\\\[10pt]
0 & \text{elsewhere}
\end{cases}
\]

The exponential distribution models the time between random independent events occurring at a constant average rate. It is defined by the following probability density function:

\[
f(x; \lambda) =
\begin{cases}
\lambda e^{-\lambda x} & \text{for } x > 0 \\\\[0.6em]
0 & \text{for } x \le 0
\end{cases}
\]