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…