BETA DISTRIBUTION HISTORY The distribution has a very long history.
The "problem in the doctrine of chances" that Bayes treated produced a beta distribution for the posterior density of the probability of a success in Bernoulli trials. In the early 20th century English literature it was usual to refer to the distribution by its designation in the Pearson family of curves. However the new text-books of the 1940s did not favour the Pearson classification and the beta designation has become standard. An Essay towards solving a Problem in the Doctrine of Chances is a work on the mathematical theory of probability by the Reverend Thomas Bayes, published in 1763, two years after its author's death. It included a statement of a special case of what is now called Bayes' theorem. In 18th-century English, the phrase "doctrine of chances" meant the theory of probability. It had been introduced as the title of a book by Abraham Demoivre. Bayes supposed a sequence of independent experiments, each having as its outcome either success or failure, the probability of success being some number p between 0 and 1. But then he supposed p to be an uncertain quantity, whose probability of being in any interval between 0 and 1 is the length of the interval. In modern terms, p would be considered a random variable uniformly distributed between 0 and 1. Conditionally on the value of p, the trials resulting in success or failure are independent, but unconditionally (or "marginally") they are not. That is because if a large number of successes are observed, then p is more likely to be large, so that success on the next trial is more probable. The question Bayes addressed was: what is the conditional probability distribution of p, given the numbers of successes and failures so far observed. The answer is that its probability density function is
(and (p) = 0 for p < 0 or p > 1) where k is the number of successes so far observed, and n is the number of trials so far observed. This is what today is called the Beta distribution with parameters k + 1 and n k + 1.
P.D.F. OR P.F. Probability density function The probability density function of the beta distribution is:
where (z) is the gamma function. The beta function, B, appears as a normalization constant to ensure that the total probability integrates to unity. A random variable X that is Beta-distributed with shape and is denoted XBe(,) Cumulative distribution function The cumulative distribution function is
where Bx(,) is the incomplete beta function and Ix(,) is the regularized incomplete beta function
PARAMETER Parameter estimation Let
be the sample mean and
be the sample variance. The method-of-moments estimates of the parameters are
When the distribution is required over an interval other than [0, 1], say
, then replace
with
and
with
in the above equations.
There is no closed-form of the maximum likelihood estimates for the parameters. Reference:
EXAMPLE OF P.F. OR P.D.F. PLOT Probability density function
Cumulative distribution function
MEAN AND VARIANCE The expected value (mean) (), variance (second central moment), skewness (third central moment), and kurtosis excess (fourth central moment) of a Beta distribution random variable X with parameters and are:
The skewness is
The kurtosis excess is:
or:
In general, the kth raw moment is given by
where (x)k is a Pochhammer symbol representing rising factorial. It can also be written in a recursive form as
One can also show that
EXAMPLE 1 Important computational aids for the numerical evaluation of incomplete integrals of gamma and beta distributions involve expressing such integrals as sums of probabilities of particular Poisson and binomial distributions. a) Prove that
where b) Prove that and
and where is a positive integer.
where and are positive integers and where SOLUTION 1 a) With , we have
which is when .
b) With
, we have
Thus, since we have
which is when .
EXAMPLE 2 Let be a random sample of binary random variables with , show that a level . Using the confidence interval for
cumulative distribution function of is
where defined to be . SOLUTION 2 Since
is the
th quantile of the F-distribution
, and
is
has the binomial distribution with size
and probability
and the binomial family has , is decreasing is the solution to
monotone likelihood ratio in , the cumulative distribution function of in for fixed . A level and confidence interval for is . Let , where
is the solution to
be a random variable having
the beta distribution with parameter
. Using integral by parts, we obtain that
Therefore,
is the if
th quantile of the beta distribution with parameter . For , it is the solution to if .Then, . Hence,
if is the
and is equal to
th quantile of the beta distribution with parameter to if . Let be a random variable having the F-distribution . Hence
and is equal has
the beta distribution with parameter
when
is defined to be . Similarly,
Note that
has the F-distribution
. Hence,
