What are valid values for alpha and beta?

Both alpha (α) and beta (β) must be strictly positive real numbers (> 0). Common choices range from 0.5 to 10. When both equal 1, the result is a uniform distribution on [0, 1].

When does the mode exist?

The mode is defined only when both α > 1 and β > 1. It equals (α − 1) / (α + β − 2). When α = β = 1 every point is a mode (uniform), and when α ≤ 1 or β ≤ 1 the mode is at a boundary or undefined.

What is the difference between PDF and CDF?

The PDF (probability density function) gives the relative likelihood of the random variable taking a specific value x. The CDF (cumulative distribution function) gives the probability that the variable is less than or equal to x, i.e., P(X ≤ x).

How is the regularized incomplete beta function computed?

This calculator uses a continued-fraction expansion (Lentz algorithm) to numerically evaluate the regularized incomplete beta function I_x(α, β), which equals the Beta CDF. This approach is accurate to about 12 significant digits for typical parameter ranges.

What does a symmetric Beta distribution look like?

When α = β the distribution is symmetric around 0.5. Small equal values (e.g., α = β = 0.5) create a U-shaped curve; equal values greater than 1 (e.g., α = β = 5) create a bell-shaped curve centered at 0.5.

Can this calculator handle very large alpha and beta values?

Yes, up to values around 1000. For extremely large parameters, the distribution becomes very narrow and approaches a normal distribution near its mean. JavaScript floating-point arithmetic limits precision beyond that range.

Beta Distribution Calculator

Enter shape parameters alpha and beta to compute PDF, CDF, mean, variance, mode, and visualize the Beta distribution curve.

Shape Parameters

Alpha (α) — must be > 0

Beta (β) — must be > 0

x value (0 to 1) — evaluate PDF & CDF

Distribution Statistics

PDF at x —

CDF at x — P(X ≤ x) —

Mean —

Variance —

Std Deviation —

Mode —

Skewness —

Kurtosis (excess) —

Key Formulas

Mean: α / (α + β)
Variance: αβ / ((α+β)²(α+β+1))
Mode: (α−1) / (α+β−2), when α,β > 1
Skewness: 2(β−α)√(α+β+1) / ((α+β+2)√(αβ))
PDF: x^(α−1)(1−x)^(β−1) / B(α,β)
CDF: I_x(α,β) — regularized incomplete beta

PDF Curve

α=2, β=5

The vertical line marks the current x value. The filled area shows P(X ≤ x).

Common Presets

Summary

Enter shape parameters alpha and beta to compute PDF, CDF, mean, variance, mode, and visualize the Beta distribution curve.

How it works

Enter the shape parameter alpha (α > 0) in the first field.
Enter the shape parameter beta (β > 0) in the second field.
Optionally enter an x value in [0, 1] to evaluate the PDF and CDF at that point.
The calculator instantly shows mean, variance, mode, and skewness.
The PDF curve chart updates automatically so you can visualize the distribution shape.

Use cases

Model the probability of success in Bayesian A/B testing.
Estimate task completion probabilities in PERT project scheduling.
Analyze proportions and rates in statistical quality control.
Model click-through rates, conversion rates, and other bounded metrics.
Perform prior and posterior distribution analysis in Bayesian inference.
Study continuous random variables constrained between 0 and 1.
Visualize how shape parameters affect distribution skewness and peakedness.
Compute exact CDF values for hypothesis testing with Beta-distributed variables.

Frequently Asked Questions

Last updated: 2026-09-22 · Reviewed by Nham Vu