Probability and Statistics II week 10

THE BETA DISTRIBUTION X is uniform on the interval [a, b], the p.d.f. of the uniform distribution is defined as: The uniform density is a special case of the beta distribution, which is defined in the following way: Definition 3.2.5. A random variable X has a beta distribution and it is referred to as a beta random variable if and only if its probability density function is given by where are the parameters of the distribution. In recent years, the beta distribution has found important applications in Bayesian inference, where the parameters are looked upon as random variables, and there is a need for a fairly “flexible” probability density for the parameter of the binomial distribution, which takes on nonzero values only on the interval from 0 to 1. By “flexible” we mean that the probability can take on a great variety of different shapes. The use of the beta distribution will be discussed later on. We shall not prove here that the total area under the curve of the beta distribution, like that of any probability density, is equal to 1, but in the proof of the theorem that follows, we shall make use of the fact that and hence that This integral defines the beta function, whose values are denoted ; in other words, , that is; Examples Detailed discussion of the beta function may be found in any textbook on advanced calculus. It is quite straight forward to evaluate the moments of a Beta random variable. Theorem 3.2.1 The mean of the Beta distribution is and the variance is Proof From If k = 1, we have