continuous distribution

The Beta distribution is a continuous probability distribution defined over the open interval \( (0, 1) \). In formal terms, the beta distribution is defined by the following probability density function:

\[
B(x; \alpha, \beta) =
\frac{x^{\alpha – 1}(1 – x)^{\beta – 1}}{B(\alpha, \beta)}
\quad 0 < x < 1 \]

The Student’s t distribution models how a sample mean behaves when the population variance is unknown and estimated from data. It is defined as:

is defined as:

\[
T = \frac{Z}{\sqrt{V/k}}
\]

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:

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