What is beta distribution?

The beta distribution is a continuous probability distribution used to model random variables that are constrained to lie between 0 and 1.

The beta distribution is defined by two parameters, α and β, both of which are positive real numbers. The probability density function (PDF) of the beta distribution is given by:

f(x; α, β) = x^(α-1) * (1-x)^(β-1) / B(α, β)

where x is the random variable, B(α, β) is the beta function, and α and β are the shape parameters. The beta function is defined as:

B(α, β) = Γ(α) * Γ(β) / Γ(α + β)

where Γ is the gamma function.

The mean and variance of the beta distribution are given by:

μ = α / (α + β)

σ^2 = αβ / [(α + β)^2 (α + β + 1)]

The beta distribution is commonly used in Bayesian statistics as a prior distribution for probabilities. It is also used in reliability analysis, queuing theory, and in modelling proportions and percentages.

In summary, the beta distribution is a versatile probability distribution that is useful for modelling random variables that are constrained to lie between 0 and 1. It is defined by two shape parameters, α and β, and has a probability density function that can be used to calculate probabilities and moments.