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The probability density function for a beta distribution with parameters a and b (a, b > 0) is of the form:B(x; a, b) = c*x*(a-1)*(1-x)(b-1) where 0 <= x <= 1.
c is a constant such that the integral is 1 and it can be shown that c = G(a+b)/[G(a)*G(b)] where G is the Gamma function.
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Is the beta distribution is a special case of dirichlet distribution?
In a sense.Beta distributions are the marginal distributions of the Dirichlet distribution.
Is the beta distribution closed under convex combinations viz is the convex combination of two or more beta distributions also a beta distribution?
No. For a convex combination of distributions, the density is also a convex combination of the individual densities and one can easilly check that the convex combination of beta densities is not again a beta density.
What is the difference between beta and normal distribution?
The probability density functions are different in shape and the domain. The domain of the beta distribution is from 0 to 1, while the normal goes from negative infinite to positive infinity. The shape of the normal is always a symmetrical, bell shape with inflection points on either sides of the mean. The beta distribution can be a variety of shapes, symmetrical half circle, inverted (cup up) half circle, or asymmetrical shapes. Normal distribution has many applications in classical hypothesis testing. Beta has many applications in Bayesian analysis. The uniform distribution is considered a specialized case of the beta distribution. See related links.
What is the formula for finding probabilities?
The formula for finding probability depends on the distribution function.
What does unimodal mean?
Please consider the probability density function graphs for the beta distribution, given in the link. For alpha=beta=2, the density is unimodal, which is to say, it has a single maximum. In contrast, for alpha=beta=0.5, the density is bimodal; it has two maxima.
