Beta Probability Distribution Calculator

Calculator Inputs

Use decimals between 0 and 1 for x, bounds, percentile, and confidence level.

Alpha (α)

Beta (β)

Evaluation Point x

Lower Bound

Upper Bound

Percentile p

Sample Size

Confidence Level

Example Data Table

Case	Alpha	Beta	x	Lower	Upper	Percentile p	Sample Size
Quality Score Prior	2.50	4.00	0.35	0.20	0.60	0.90	100
Balanced Belief	5.00	5.00	0.50	0.40	0.70	0.95	250
Boundary Favoring Low Values	0.80	3.20	0.20	0.05	0.35	0.80	80
Boundary Favoring High Values	4.50	0.90	0.85	0.70	0.98	0.75	120

Formula Used

Probability Density Function: f(x) = x^(α-1) × (1-x)^(β-1) / B(α,β), for 0 ≤ x ≤ 1.

Beta Function: B(α,β) = Γ(α)Γ(β) / Γ(α+β).

Cumulative Distribution: F(x) = Iₓ(α,β), the regularized incomplete beta function.

Interval Probability: P(L ≤ X ≤ U) = F(U) - F(L).

Mean: α / (α + β).

Variance: αβ / [(α+β)²(α+β+1)].

Mode: (α-1) / (α+β-2) when α > 1 and β > 1.

Quantile: the calculator finds the x-value whose cumulative probability matches your chosen percentile.

How to Use This Calculator

Enter positive values for alpha and beta to define the distribution shape.
Provide a point x between 0 and 1 for density and cumulative probability.
Set lower and upper bounds to evaluate probability over an interval.
Enter percentile and confidence level as decimals, such as 0.90 or 0.95.
Choose a sample size if you want an expected count inside the interval.
Click the calculate button to show results, graph, and export options above the form.

Frequently Asked Questions

1. What does the beta distribution model?

It models uncertain values restricted to the interval from 0 to 1. Common uses include probabilities, conversion rates, reliability proportions, and Bayesian posterior distributions for binary outcomes.

2. Why must alpha and beta be positive?

Positive shape parameters keep the beta function valid and produce a proper probability distribution. Zero or negative values break the density formula and the related moment calculations.

3. What does the PDF value mean here?

The PDF gives the density at one exact point. It shows relative concentration, not direct probability at a single exact x. Interval probabilities come from the CDF difference.

4. How should I read the CDF output?

The CDF at x means the probability that the random variable is less than or equal to that x-value. It always increases from 0 to 1.

5. Why is the mode sometimes unavailable?

An interior mode exists only when alpha and beta are both greater than 1. Otherwise, the distribution may peak at a boundary or remain flat, so the standard interior mode formula does not apply.

6. What does the percentile output represent?

It is the x-value where the cumulative probability reaches your chosen percentile. For example, a 0.90 percentile means 90 percent of the distribution lies at or below that value.

7. What is the expected count in interval?

The calculator multiplies the interval probability by your sample size. This gives the expected number of observations that would fall inside the selected range on average.

8. Why can the graph spike near 0 or 1?

When alpha or beta is below 1, the beta density can become very steep near a boundary. That behavior is valid and reflects strong concentration close to 0 or 1.