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The beta pdf is

where *B*( · ) is the Beta function.
The indicator function *I*_{(0,1)}(*x*)
ensures that only values of *x* in the range (0
1) have nonzero probability.

The beta distribution describes a family of curves that are unique in that they are nonzero only on the interval (0 1). A more general version of the function assigns parameters to the endpoints of the interval.

The beta cdf is the same as the incomplete beta function.

The beta distribution has a functional relationship with the *t* distribution.
If *Y* is an observation from Student's *t* distribution
with *ν* degrees of freedom, then the following
transformation generates *X*, which is beta distributed.

If *Y*~*t*(*v*),
then

This relationship is used to compute values of the *t* cdf
and inverse function as well as generating *t* distributed
random numbers.

Suppose you are collecting data that has hard lower and upper bounds of zero and one respectively. Parameter estimation is the process of determining the parameters of the beta distribution that fit this data best in some sense.

One popular criterion of goodness is to maximize the likelihood
function. The likelihood has the same form as the beta pdf. But for
the pdf, the parameters are known constants and the variable is *x*. The likelihood function reverses
the roles of the variables. Here, the sample values (the *x*'s)
are already observed. So they are the fixed constants. The variables
are the unknown parameters. Maximum likelihood estimation (MLE) involves
calculating the values of the parameters that give the highest likelihood
given the particular set of data.

The function `betafit` returns
the MLEs and confidence intervals for the parameters of the beta distribution.
Here is an example using random numbers from the beta distribution
with `a = 5` and`b = 0.2`.

rng('default'); % For reproducibility r = betarnd(5,0.2,100,1); [phat, pci] = betafit(r)

phat = 7.4911 0.2135 pci = 5.0861 0.1744 11.0334 0.2614

The MLE for parameter `a` is 7.4911, compared
to the true value of 5. The 95% confidence interval for `a` goes
from 2.8051 to 6.2610, which does not include the true value. While
this is an unlikely result, it does sometimes happen when estimating
distribution parameters.

Similarly the MLE for parameter `b` is 0.2135,
compared to the true value of 0.2. The 95% confidence interval for `b` goes
from 0.1771 to 0.2832, which does include the true value. In this
made-up example you know the "true value." In experimentation
you do not.

The shape of the beta distribution is quite variable depending on the values of the parameters, as illustrated by the plot below.

X = 0:.01:1; y1 = betapdf(X,0.75,0.75); y2 = betapdf(X,1,1); y3 = betapdf(X,4,4); figure; plot(X,y1,'Color','r','LineWidth',2); hold on; plot(X,y2,'LineStyle','-.','Color','b','LineWidth',2); plot(X,y3,'LineStyle',':','Color','g','LineWidth',2); legend({'a = b = 0.75','a = b = 1','a = b = 4'},'Location','NorthEast'); hold off;

The constant pdf (the flat line) shows that the standard uniform
distribution is a special case of the beta distribution, which occurs
when `a = b = 1`.

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