Statistics & Probability

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Beta Distribution Calculator.

Evaluate the beta probability density and moments.

On-device calculationNo signup
01

Set your values

Results update as you type.

probabilityDensity: 0.9375

probabilityDensity

0.937500
mean: 0.285714

mean

0.285714
variance: 0.02551

variance

0.025510

Results update automatically as you type.

Use Cases

Modeling probabilities and proportions

Use the beta distribution to model random variables that represent probabilities or proportions, such as conversion rates, success rates, or the distribution of a percentage.

Example: Estimate the density of a conversion rate of 0.3 with alpha=2 and beta=5.

Bayesian prior/posterior analysis

In Bayesian statistics, the beta distribution is a conjugate prior for the binomial distribution. Use this calculator to compute the density and moments of a prior or posterior distribution.

Example: Compute the mean and variance of a posterior beta(10, 20) distribution.

Frequently Asked Questions

What does the beta distribution calculator compute?
It evaluates the probability density function (PDF) at a given x (between 0 and 1) and calculates the moments: mean, variance, skewness, and kurtosis, based on the alpha and beta shape parameters you input.
What are the alpha and beta parameters?
Alpha and beta are positive shape parameters that determine the shape of the beta distribution. They control the skewness and concentration of the distribution. For example, alpha=beta=1 gives a uniform distribution.
Can x be outside 0 to 1?
No, the beta distribution is defined only for x in the interval [0, 1]. The calculator expects x to be between 0 and 1 inclusive. Values outside this range are not valid.

Tips & Common Mistakes

Tips

  • Ensure alpha and beta are positive numbers; otherwise, the beta distribution is not defined.
  • For a uniform distribution, set alpha=1 and beta=1.
  • The mean of the beta distribution is alpha/(alpha+beta). Use this to quickly check your results.
  • If you need the cumulative probability, note that this calculator only provides the density and moments, not the CDF.

Common Mistakes to Avoid

  • Entering x outside the range [0,1] – the beta distribution is only defined for x between 0 and 1.
  • Using non-positive values for alpha or beta – they must be greater than 0.
  • Confusing the density value with probability – the PDF value is not a probability itself; it can be greater than 1.

Last updated: August 13, 2026