Beta Distribution Calculator

The beta distribution is governed by two shape parameters, denoted as alpha (a) and beta (b), which determine the skewness and curvature of the probability curve. The expected value or mean is calculated using the formula
Mean = a / (a + b).
The dispersion of the data is measured by the variance, computed via
Variance = (a * b) / ((a + b)^2 * (a + b + 1)).
To find the standard deviation, we simply take the square root of the variance. Furthermore, the calculator uses the beta function and regularized incomplete beta functions to evaluate the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) at any specific value of x.

Imagine you are analyzing an A/B test for an e-commerce checkout page. Based on prior traffic, your success and failure counts correspond to shape parameters alpha = 8 and beta = 2. Let us find the mean, variance, and probability density at x = 0.75.
1. Calculate the mean: Mean = 8 / (8 + 2) = 8 / 10 = 0.80.
2. Calculate the variance: Variance = (8 * 2) / ((8 + 2)^2 * (8 + 2 + 1)) = 16 / (100 * 11) = 16 / 1100 ≈ 0.0145.
3. Calculate the standard deviation: SD = sqrt(0.0145) ≈ 0.1206.
4. Evaluate the PDF at x = 0.75 using the beta function formulation to find the density value at that exact conversion rate point. This gives researchers a clear statistical profile of the underlying conversion likelihood.

When selecting shape parameters, remember that if alpha equals beta, the distribution is perfectly symmetrical. If alpha is less than beta, the distribution is skewed to the right, whereas alpha greater than beta yields left-skewness. Always ensure your input variable x falls strictly between 0 and 1, as values outside this range are undefined for standard beta distributions.