Beta distribution is commonly used as a prior distribution in Bayesian statistics, particularly for modeling probabilities that are bounded between 0 and 1. It is characterized by two parameters, alpha and beta, which control the shape of the distribution.
1. INTUITION
One way to think about the beta distribution is to consider it as a way of modeling the distribution of probabilities.
For example, let’s say we are flipping a coin and we want to model the probability of getting heads. We know that the probability of getting heads is between 0 and 1, so we can use a beta distribution to model this probability.
The beta distribution has a probability density function (PDF) given by:
where x is a value between 0 and 1, alpha and beta are the shape parameters, and B(alpha, beta) is the beta function, which is used to normalize the PDF. The beta function is defined as:
where Gamma is the gamma function, which is a generalization of the factorial function to non-integer values.
2. CONNECTION WITH BINOMIAL DISTRIBUTION
The beta distribution is intimately connected to the binomial distribution. Specifically, if we have a binomial distribution with parameters n…
