Beta Distribution Calculator
Calculate P(X < x) for the beta distribution.
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Enter the alpha and beta shape parameters and an x value between 0 and 1. The calculator computes the cumulative distribution function P(X < x) using numerical integration. It also shows the mean, mode, and standard deviation of the distribution.
About This Calculator
The beta distribution is a flexible distribution on the interval [0, 1], commonly used as a prior distribution in Bayesian statistics. Its shape is determined by two parameters, alpha and beta. When alpha = beta = 1, it reduces to the uniform distribution. It is used to model proportions, probabilities, and rates.
Frequently Asked Questions
What do alpha and beta control?
Alpha controls the weight toward 1, beta controls the weight toward 0. When alpha > beta, the distribution skews right. When both are large, the distribution becomes concentrated around the mean.
Why is the beta distribution used in Bayesian statistics?
Because it is the conjugate prior for the binomial distribution. Starting with a Beta(alpha, beta) prior and observing k successes in n trials, the posterior is Beta(alpha+k, beta+n-k).
What is the mean of the beta distribution?
The mean is alpha / (alpha + beta). The mode (for alpha,beta > 1) is (alpha-1) / (alpha+beta-2).