Beta distribution

Occasionally variables can only take values within a restricted range. For example, the concentration of a mineral in rocks must be between zero and one.

The family of beta distributions is flexible enough to model many such variables.

Definition

A random variable \(X\) is said to have a Beta distribution with parameters \(\alpha \gt 0\) and \(\beta \gt 0\),

\[ X \;\;\sim\;\; \BetaDistn(\alpha,\; \beta) \]

if its probability density function is

\[ f(x) \;\;=\;\; \begin{cases} \dfrac {\Gamma(\alpha +\beta) }{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}& \text{if }0 \lt x \le 1 \\[0.4em] 0 & \text{otherwise} \end{cases} \]

A special case of the beta distribution arises when \(\alpha = \beta = 1\):

\[ \BetaDistn(\alpha = 1,\; \beta = 1) \;\;\equiv\;\; \RectDistn(0, 1) \]

Larger values of the parameters decrease the spread of the distribution, whereas smaller values "push the distribution towards zero and one".

Shape of the Beta distribution

Rather than directly adjusting the values of the two Beta distribution parameters, \(\alpha\) and \(\beta\), we will illustrate the possible shapes of the distribution with sliders for its mean and standard deviation.

With the distribution's mean left at 0.5, use the lower slider to alter the standard deviation and observe that:

Reduce the mean to 0.2 then again adjust the standard deviation, observing that high standard deviation again corresponds to values that are likely to be near zero or one. (When the standard deviation is very high, the probability densities at zero and one are actually both infinite, with considerably greater area near zero than one.)