4.5.3 The Beta Distribution

All families of continuous distributions discussed so far except for the uniform distribution have positive density over an infinite interval (though typically the density function decreases rapidly to zero beyond a few standard deviations from the mean). The beta distribution provides positive density only for $X$ in an interval of finite length.

Definition

A random variable $X$ is said to have a beta distribution with parameters $\alpha ,\beta$ (both positive), $A$ , and $B$ if the pdf of $X$ is

$$f\left(x;\alpha ,\beta ,A,B\right)=\{\begin{array}{cc}\frac{1}{B-A}\cdot \frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\cdot \Gamma \left(\beta \right)}{\left(\frac{x-A}{B-A}\right)}^{\alpha -1}{\left(\frac{B-x}{B-A}\right)}^{\beta -1}& A\le x\le B\\ 0& \text{ otherwise }\end{array}$$

The case $A=0,B=1$ gives the standard beta distribution.

Figure 4.32 illustrates several standard beta pdf’s. Graphs of the general pdf are similar, except they are shifted and then stretched or compressed to fit over $\left[A,B\right]$ . Unless $\alpha$ and $\beta$ are integers, integration of the pdf to calculate probabilities is difficult. Either a table of the incomplete beta function or appropriate software should be used. The mean and variance of $X$ are

$$\mu =A+\left(B-A\right)\cdot \frac{\alpha }{\alpha +\beta }$$ $${\sigma }^2=\frac{{\left(B-A\right)}^2\alpha \beta }{{\left(\alpha +\beta \right)}^2\left(\alpha +\beta +1\right)}$$

Figure 4.32 Standard beta density curves

EX 4.28

The standard beta distribution is commonly used to model variation in the proportion or percentage of a quantity occurring in different samples, such as the proportion of a 24-hour day that an individual is asleep or the proportion of a certain element in a chemical compound.