Free Beta Distribution Calculator

Mode

Parameter α

Parameter β

x

Enter α, β, and x to compute beta distribution results.

The beta distribution calculator described here is a specialized probability distribution calculator that focuses on the beta family — one of the most flexible continuous distribution families used in statistics. It generates graphical outputs for both the probability density function (PDF) and the cumulative distribution function (CDF) of a user‑specified beta distribution, and it also computes key quantities such as probabilities, mean, variance, skewness, and more. Whether you are new to beta distributions or need quick numerical results, this tool provides an interactive way to explore how the shape parameters α and β influence the behavior of the distribution.

What Is the Beta Distribution?

The beta distribution defines a family of continuous probability distributions whose support is the interval $[0,1]$ . What makes this family so valuable is the extraordinary range of shapes it can assume: depending on the two positive parameters α and β (called shape parameters), the density can be symmetric, skewed to the left or right, unimodal, bimodal, U‑shaped, or even a straight line. This flexibility makes it a natural model for proportions, probabilities, and other quantities that are bounded between 0 and 1.

PDF and CDF Formulas

The probability density function (PDF) of a beta distribution is given by:

f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)}, \qquad 0 \le x \le 1,\ \alpha>0,\ \beta>0,

where the normalizing constant $B(\alpha,\beta)$ is the beta function, which can be expressed in terms of the gamma function:

B(\alpha,\beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}.

The cumulative distribution function (CDF) and its inverse (the quantile function) do not have simple closed‑form expressions; they are evaluated numerically by the calculator.

How to Use This Calculator

Using the tool is straightforward:

Choose a mode: Probability calculator, Sample generator, PDF, CDF, Quantile function, or Common measures.
Enter the shape parameters α and β (both > 0).
For probability, PDF, CDF, or quantile modes, provide the value of the argument x between 0 and 1.
For sample generation, specify how many random numbers you want.
The results — graphs and numerical values — are displayed instantly.

Typical Shape Examples

The beta distribution can take many forms. Below are representative combinations of α and β.

Symmetric (α = β)

α = β = 0.5 → U‑shaped density (peaks near 0 and 1)
α = β = 1 → standard uniform distribution
α = β = 2 → parabolic (bell‑like) shape
α = β = 6 → bell‑shaped, approaching normal
α = β → ∞ → the distribution collapses to a point at x = 0.5 (Dirac delta)

Skewed (α ≠ β)

α = 0.7, β = 0.8 → mildly U‑shaped with right skew
α = 2, β = 5 → right‑skewed unimodal
α = 0.5, β = 2 → J‑shaped (decreasing)
α = 1, β = 2 → downward‑sloping straight line
α = 1, β = 5 → strictly decreasing

If α < β, the distribution is right‑tailed (positive skewness); swapping the two parameters yields the mirror‑image left‑tailed distribution.

Key Statistical Measures

Several important characteristics of a beta distribution can be computed directly from α and β:

Mean (expected value)

\mu = \frac{\alpha}{\alpha+\beta}

Variance

\sigma^{2} = \frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}

Skewness

\gamma_{1} = \frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}

Mode (when $\alpha>1$ and $\beta>1$ )

\text{Mode} = \frac{\alpha-1}{\alpha+\beta-2}

The median does not have a simple closed form, except in the symmetric case (α = β) where it is 0.5.

Role in Bayesian Inference

The beta distribution is a popular choice for prior distributions in Bayesian statistics because it is the conjugate prior for several likelihood functions — specifically the binomial, negative binomial, and geometric distributions. When a beta prior is used with one of these likelihoods, the posterior distribution is also a beta distribution, and updating only involves adding the number of successes to α and the number of failures (or trials, depending on the case) to β. This property eliminates the need for costly numerical integration when computing the posterior.

FAQ

1. How do I use this calculator to find P(X ≤ x) for a given beta distribution?

Select the Cumulative density function (CDF) mode, enter the shape parameters α and β, and provide the value of x (between 0 and 1). The calculator will return the cumulative probability P(X ≤ x).

2. What happens to the shape of the beta distribution when α = β?

When the two shape parameters are equal, the distribution is symmetric about x = 0.5. For small values (e.g., α=β=0.5) it is U‑shaped; for α=β=1 it is uniform; for larger equal values (e.g., α=β=6) it becomes bell‑shaped and approaches a normal distribution.

3. Can I generate random numbers that follow a beta distribution with this tool?

Yes. Switch to the Sample generator mode, specify the values of α and β, and choose how many random numbers you need. The calculator will output a set of simulated values from that beta distribution.

4. How are the mean and variance of a beta distribution calculated?

Given shape parameters α and β, the mean is α/(α+β) and the variance is (αβ)/((α+β)²(α+β+1)). These formulas are implemented in the Common measures mode of the calculator.

5. Why is the beta distribution so often used as a prior in Bayesian analysis?

The beta distribution is the conjugate prior for binomial, negative binomial, and geometric likelihoods. This means that when you use a beta prior, the posterior distribution is also a beta — you simply update the parameters by adding the number of successes to α and failures to β, avoiding complex numerical calculations.

How to Use

Select the calculation mode from the dropdown.
Enter the shape parameters α and β (positive numbers).
View the computed results instantly.