Enter shape parameters α (alpha) and β (beta) to get the full picture: PDF value, CDF probability, mean, variance, mode, skewness, and shape classification. Three probability modes supported — P(X ≤ x), P(X ≥ x), and P(a ≤ X ≤ b). Also includes a Bayesian posterior update calculator.
✓Formulas verified: Wikipedia Beta Distribution + NIST Digital Library of Mathematical Functions —
📐 Shape Parameters & Probability Type
> 0
Controls weight toward 1. α > 1 creates interior mode. Try α=2, β=5.Alpha must be greater than 0.
> 0
Controls weight toward 0. Equal α=β gives symmetric distribution.Beta must be greater than 0.
Choose how to compute probability for a given x value.
[0,1]
Enter a value between 0 and 1 for x.x must be between 0 and 1.
[0,1]
Upper bound b for P(a ≤ X ≤ b). Must be > a.b must be between 0 and 1, and greater than a.
Probability Result
0.0000
Beta(α, β) probability
🧠 Bayesian Posterior Update — Observe s Successes in n Trials
Prior
→
Posterior
⚠️ Note: CDF values are computed via numerical integration (Simpson’s rule, 1000 steps). Results are highly accurate but may show small floating-point rounding for extreme parameter values. For research use, verify with a statistical software package such as R or Python scipy.stats.
Comprehensive reference for all beta distribution formulas used in this calculator: PDF, mean, variance, mode, skewness, and the conjugate prior property. Verified PDF formula f(x;α,β) = x^(α-1)(1-x)^(β-1)/B(α,β).
Authoritative source for the regularized incomplete beta function I_x(α,β) used to compute the CDF. The NIST DLMF is the standard reference for special mathematical functions.
Beta Distribution — PDF, CDF, Formulas, and Bayesian Applications
The beta distribution is one of the most versatile tools in statistics. It lives entirely on the interval [0, 1], which makes it the natural choice any time your quantity of interest is a probability, a proportion, or a percentage. Want to model the true conversion rate of a marketing campaign? The defect probability of a manufacturing process? The uncertainty in a project’s completion time? The beta distribution handles all of these, and it does so with remarkable flexibility — the same family of distributions can produce shapes that are uniform, bell-shaped, J-shaped, U-shaped, or heavily skewed, simply by adjusting two parameters.
What Is the Beta Distribution and How Does It Work?
The beta distribution is parameterized by two positive real numbers, α (alpha) and β (beta), called shape parameters. Together they determine everything about the distribution’s shape, center, and spread. The distribution is defined on [0, 1], making it ideal for quantities that are inherently bounded. The probability density function (PDF) tells you the relative likelihood density at each value of x between 0 and 1, while the cumulative distribution function (CDF) gives the total probability that a beta-distributed variable falls at or below a given value.
Here’s the key insight that makes α and β intuitive: in a Bayesian context for modeling a success probability, α tracks the number of prior successes (plus 1) and β tracks the number of prior failures (plus 1). Starting from Beta(1, 1) — which is the uniform distribution representing total ignorance — every success you observe increments α and every failure increments β. The distribution narrows and shifts as evidence accumulates, always remaining a beta distribution.
Beta Distribution Formulas — PDF, Mean, Variance, Mode and Skewness
🧮 Beta Distribution Key Formulas (All Verified)
PDF: f(x; α, β) = x^(α−1) × (1−x)^(β−1) / B(α, β)
B(α, β) = Γ(α) × Γ(β) / Γ(α+β) (beta function)
Mean: μ = α / (α + β)
Variance: σ² = (αβ) / [(α+β)² × (α+β+1)]
Mode: (α−1) / (α+β−2) when α>1 and β>1
Skewness: 2(β−α)√(α+β+1) / [(α+β+2)√(αβ)]
Source: Wikipedia Beta Distribution | NIST DLMF 8.17. Worked example: α=2, β=5 → mean=0.2857, variance=0.02551, mode=0.2000, skewness=+0.596 (right-skewed). Confirmed with Python scipy.stats.beta.
Shape Parameter Guide — What Alpha and Beta Actually Control
The beauty of the beta distribution is that you can engineer almost any shape you need by setting α and β appropriately. This table summarizes how the distribution behaves across key parameter combinations — something most competitors explain poorly:
Alpha (α)
Beta (β)
Distribution Shape
Real-World Use Case
= 1
= 1
Uniform [0,1]
Complete ignorance prior (no preference)
> 1
> 1, equal
Symmetric bell curve
Symmetric belief centered at 0.5 (e.g., fair coin)
> β
< α
Left-skewed toward 1
High success rate product, conversion rate near 100%
< β
> α
Right-skewed toward 0
Low defect rate, rare event probability
< 1
< 1
U-shaped, poles at 0 and 1
Polarizing outcomes (all-or-nothing scenarios)
< 1
≥ 1
J-shaped, peak at 0
Modeling very low probability events
≥ 1
< 1
Reverse J, peak at 1
Modeling high-confidence success scenarios
Large
Large, equal
Tight bell near 0.5
High-evidence posterior with many trials observed
Beta Distribution in Bayesian Statistics — Conjugate Prior for the Binomial
The single most important property of the beta distribution in applied statistics is that it is the conjugate prior for the binomial likelihood. What this means in practice: if you express your prior belief about a probability p as Beta(α, β), and you then observe s successes in n trials, your updated posterior belief is simply Beta(α + s, β + n − s). No complex integration required. The posterior is still a beta distribution.
This is why beta is so widely used in A/B testing, conversion rate optimization, and any scenario where you are updating your belief about a probability based on evidence. Start with an uninformative prior Beta(1,1), observe 30 conversions in 200 visits, and your posterior is Beta(31,171) — giving a mean of 31/202 = 15.3% and a tight variance reflecting your 200-observation sample size.
💡 PERT Project Management: The beta distribution also underlies the PERT technique for estimating task durations. Given optimistic (a), most likely (m), and pessimistic (b) estimates, the expected task duration is (a + 4m + b) / 6 — the mean of a beta distribution fitted to [a, b] with mode at m. PERT deliberately uses the beta distribution’s asymmetric flexibility to reflect the reality that projects almost always take longer than planned, not shorter.
Frequently Asked Questions
The beta distribution is a continuous probability distribution defined on [0, 1], parameterized by two positive shape parameters α and β. It models quantities that are inherently proportions or probabilities — things that must fall between 0 and 1. Its flexibility is remarkable: depending on α and β, it can be uniform, bell-shaped, J-shaped, U-shaped, or skewed in either direction. This makes it the standard choice in Bayesian statistics, quality control, PERT project management, and A/B testing.
Mean = α / (α + β). For example, with α=3 and β=7: mean = 3/10 = 0.30. The mean always lies strictly between 0 and 1. When α equals β, the mean is exactly 0.5. The mean increases as α grows relative to β, shifting toward 1. Increasing both parameters proportionally leaves the mean unchanged but shrinks the variance, concentrating the distribution around its center.
α and β are both shape parameters, but they push the distribution in opposite directions. α controls the weight of the distribution near 1 — larger α shifts probability mass toward higher values. β controls weight near 0 — larger β shifts mass toward lower values. When α equals β, the distribution is symmetric around 0.5. In a Bayesian context: α represents prior successes + 1, and β represents prior failures + 1. Beta(1,1) is uniform (no prior information).
In Bayesian A/B testing, the beta distribution models the conversion rate of each variant. Start with a uniform prior Beta(1,1) for each variant. As visitors are assigned and conversions recorded, update: posterior = Beta(1 + conversions, 1 + non-conversions). Compare the posterior distributions of variant A and variant B to determine which has a higher conversion rate with what probability. This approach naturally handles small sample sizes and quantifies uncertainty, unlike frequentist p-values.
Mode = (α−1) / (α+β−2) when both α>1 and β>1. With α=3 and β=5: mode = 2/6 = 0.333. Special cases: if α=1 and β=1, the distribution is uniform (no unique mode). If α<1 or β<1, the distribution has poles at 0 or 1. If both are less than 1, the distribution is U-shaped with modes at both endpoints. If α=1 and β>1, mode is at 0. If α>1 and β=1, mode is at 1.
P(a ≤ X ≤ b) = CDF(b) − CDF(a) = I_b(α,β) − I_a(α,β), where I_x is the regularized incomplete beta function. For example, P(0.2 ≤ X ≤ 0.6) with α=2, β=5: compute CDF at 0.6 minus CDF at 0.2. This calculator does this automatically. Use range probability when you want to know the probability that a beta-distributed variable falls within a specific interval.
The CDF of the beta distribution is F(x) = I_x(α,β), the regularized incomplete beta function. It gives P(X ≤ x). There is no closed-form expression for most (α,β) pairs, so it is computed numerically. Methods include: continued fraction expansions (fast, accurate), series expansions, and numerical integration. This calculator uses numerical integration via Simpson’s rule with 1000 steps, which is accurate to at least 5 decimal places for typical parameter values.
Skewness measures how asymmetric the distribution is. Skewness = 2(β−α)√(α+β+1) / [(α+β+2)√(αβ)]. When α is less than β, skewness is positive (right-skewed): the right tail is longer and most probability mass is near 0. When α is greater than β, skewness is negative (left-skewed): mass near 1, tail on the left. When α equals β, skewness is 0 (symmetric). Right-skewed beta distributions are common in modeling defect rates, rare event probabilities, and early-stage A/B tests where most outcomes are negative.
PERT (Program Evaluation and Review Technique) uses the beta distribution to model task duration uncertainty in project management. A task is characterized by three estimates: optimistic (a), most likely (m), and pessimistic (b). The expected duration is (a + 4m + b)/6, which is the mean of a scaled beta distribution over [a, b] with mode at m. PERT chooses beta because it allows asymmetric distributions — tasks almost always run longer than planned, creating a right-skewed distribution. The standard deviation is (b−a)/6.
A conjugate prior is a prior distribution that, when combined with a particular likelihood function via Bayes’ theorem, produces a posterior in the same distributional family. Beta is conjugate to the binomial: if your prior is Beta(α,β) and your data is binomial (s successes, f failures), your posterior is Beta(α+s, β+f). This mathematical convenience means you never have to compute a normalizing constant or do numerical integration for the posterior — you just add counts. This is why beta is the default prior for modeling proportions, conversion rates, and probability parameters.
Beta(1,1) is exactly the uniform distribution on [0,1]. Every point between 0 and 1 has the same probability density. In Bayesian terms, this is the uninformative prior — it says you have no prior knowledge about the true probability. As you add evidence (incrementing α with successes and β with failures), the distribution moves away from uniform and concentrates around the observed proportion. This elegant property makes Beta(1,1) the natural starting point for any Bayesian analysis of a proportion.
Concrete examples: (1) Conversion rate modeling — tracking the posterior distribution of a website conversion rate from Beta(1,1) prior, updating with observed clicks and conversions. (2) Clinical trial success probability — modeling probability of drug efficacy before and after patient trials. (3) Election forecasting — modeling the true support proportion for a candidate with polling uncertainty. (4) Quality control — modeling defect probability per batch. (5) Project management PERT — modeling task duration distribution. (6) Insurance — modeling claim probability for a policyholder class.