What is the beta distribution and when is it used?

The beta distribution is a continuous distribution on [0, 1] with shape parameters α and β. It models proportions, probabilities, and rates. Common applications include Bayesian conjugate priors for binomial data, PERT project duration estimates, A/B test conversion rates, and quality control defect rates.

What is the PDF formula for the beta distribution?

The PDF is f(x; α, β) = x^(α-1) * (1-x)^(β-1) / B(α, β), where B is the beta function equal to Γ(α)*Γ(β)/Γ(α+β). The PDF gives density at a specific point and can exceed 1 for narrow distributions.

How is the CDF of the beta distribution computed?

The CDF equals the regularized incomplete beta function I_x(α, β), computed via Lentz continued fraction algorithm. This calculator uses the same numerically stable method as Numerical Recipes. For Beta(2, 5) at x = 0.3, the CDF is approximately 57.98%.

What is the mean of the beta distribution?

The mean is α / (α + β). For Beta(2, 5) the mean is 2/7, about 0.2857. For Beta(5, 2) it is 5/7, about 0.7143. Increasing α relative to β shifts the distribution toward higher x values.

What is the variance of the beta distribution?

The variance is α*β / ((α+β)^2 * (α+β+1)). For Beta(2,5) the variance is 10/392, about 0.0255. Larger total α+β concentrates the distribution even when the mean stays fixed.

What is the mode of the beta distribution?

For α > 1 and β > 1, the mode is (α-1)/(α+β-2). For α ≤ 1 and β > 1, the mode is 0. For α > 1 and β ≤ 1, the mode is 1. When both are ≤ 1, the distribution is U-shaped with no interior mode.

When is the beta distribution symmetric?

The beta distribution is symmetric about x = 0.5 when α equals β. Skewness is zero in this case, and mean, median, and mode all equal 0.5. The distribution becomes more concentrated around 0.5 as α = β increases.

Beta Distribution Calculator

Q: What do the shape parameters alpha and beta control?

Alpha controls weight near x = 1, and beta controls weight near x = 0. When both exceed 1, the distribution is bell-shaped. When both equal 1, it is uniform. When both are below 1, it becomes U-shaped with mass near the endpoints 0 and 1.

Q: How is the beta distribution used in Bayesian statistics?

The beta distribution is the conjugate prior for the binomial likelihood. If you start with Beta(α, β) and observe s successes and f failures, the posterior is Beta(α+s, β+f). This allows clean analytical updating without numerical integration.

Q: What is the relationship between Beta(1,1) and the uniform distribution?

Beta(1, 1) is exactly the Uniform(0, 1) distribution. The PDF is constant at 1 over [0, 1], the CDF is simply x, and every interval of the same length has the same probability. Setting α = β = 1 in this calculator confirms all these properties.

Find PDF, CDF, mean, variance, mode, median, skewness, and kurtosis for any Beta(α, β) distribution on [0, 1].

📊 What is the Beta Distribution Calculator?

The beta distribution is a continuous probability distribution defined on the closed interval [0, 1] and parameterized by two positive shape parameters, alpha and beta. Because its support is bounded between 0 and 1, it is a natural choice for modeling quantities that are themselves proportions, probabilities, or rates. This calculator handles both probability queries (what is the chance that a Beta-distributed variable falls below a given x?) and full distribution characterization (what are the mean, variance, mode, median, skewness, and kurtosis?).

The applications of the beta distribution span many disciplines. In Bayesian statistics, the beta distribution is the conjugate prior for the binomial likelihood. If a coin has an unknown bias modeled as Beta(alpha, beta) and you observe s heads and f tails, the posterior distribution is Beta(alpha + s, beta + f). This clean update rule makes the beta distribution foundational in Bayesian inference. In project management, the PERT technique uses a beta distribution to model the time needed to complete a task, where optimistic, most likely, and pessimistic estimates define the shape. In A/B testing, conversion rates for two variants are each modeled as beta distributions, and the probability that one variant is better than the other is computed analytically. In quality control, defect rates per lot are bounded in [0, 1] and often follow a beta distribution.

The shape of the distribution changes dramatically with alpha and beta. When both parameters exceed 1, the density is bell-shaped with a single interior peak. When both equal 1, the result is a flat uniform distribution. When alpha equals beta, the distribution is symmetric about 0.5. When both parameters fall below 1, the density becomes U-shaped, with large probability mass near the boundaries 0 and 1. When alpha is much larger than beta, the density concentrates near 1; when beta dominates, it concentrates near 0.

A common misunderstanding is that the PDF value represents a probability. For continuous distributions, the PDF f(x) is a probability density, not a probability. It can exceed 1 for narrow, concentrated distributions (for example, Beta(10, 10) has a PDF value above 3 near x = 0.5). The probability of falling in an interval [a, b] is the integral of f(x) from a to b, which this calculator computes as the difference of CDF values.

📐 Formulas

PDF: f(x; α, β) = x^α−1 × (1−x)^β−1 ÷ B(α, β)

α = first shape parameter (must be > 0)

β = second shape parameter (must be > 0)

B(α, β) = beta function = Γ(α) × Γ(β) ÷ Γ(α + β)

Example: Beta(2, 5), x = 0.3: f(x) = 0.3¹ × 0.7⁴ × 30 ≈ 2.161

CDF: P(X ≤ x) = I_x(α, β) = regularized incomplete beta function

Computed via Lentz continued fraction algorithm for numerical stability

Example: Beta(2, 5), x = 0.3: P(X ≤ 0.3) ≈ 57.98%

Mean = α ÷ (α + β)

Variance = αβ ÷ [(α + β)² × (α + β + 1)]

Std Dev = √Variance

Mode = (α − 1) ÷ (α + β − 2) for α > 1 and β > 1

Skewness = 2(β − α) × √(α+β+1) ÷ [(α+β+2) × √(αβ)]

Excess Kurtosis = 6[(α−β)²(α+β+1) − αβ(α+β+2)] ÷ [αβ(α+β+2)(α+β+3)]

Example: Beta(2, 5): Mean = 2/7 ≈ 0.2857, Variance ≈ 0.02551, SD ≈ 0.1597, Mode = 0.2

📖 How to Use This Calculator

Steps

Enter shape parameters - Type alpha and beta. Both must be positive. Use the sliders for quick exploration or type precise decimal values for exact work. Small values below 1 give U-shaped distributions; values above 1 give bell-shaped or skewed distributions.

Choose a mode - Click "Probability" to compute PDF, CDF P(X ≤ x), and survival P(X > x) for a specific x. Click "Distribution Stats" to see the full parameter set: mean, variance, SD, mode, median, skewness, and excess kurtosis.

Enter x and read results - In Probability mode, type the query value x between 0 and 1. The CDF, survival probability, PDF density, mean, variance, SD, and mode appear instantly. For repeated queries with different x values, only the x field needs to change.

💡 Example Calculations

Example 1 - Bayesian Posterior: Beta(2, 5)

A quality analyst models a defect rate as Beta(2, 5). What is the probability the rate is below 30%? What is the expected rate?

Set alpha = 2, beta = 5. The distribution has mean = 2/7 ≈ 0.2857 and mode = (2-1)/(7-2) = 1/5 = 0.2. The distribution is right-skewed since alpha < beta.

Compute the CDF at x = 0.3. Using the regularized incomplete beta function: P(X ≤ 0.3) = I_{0.3}(2, 5) ≈ 57.98%. So just over half the probability mass lies below 0.3.

PDF at x = 0.3: f(0.3) = 0.3^1 × 0.7^4 / B(2,5) = 0.3 × 0.2401 × 30 ≈ 2.161. Variance = 10/392 ≈ 0.02551. SD ≈ 0.1597.

P(X ≤ 0.3) ≈ 57.98%, Expected rate ≈ 0.2857

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Example 2 - Uniform Special Case: Beta(1, 1)

A researcher uses Beta(1, 1) as a non-informative prior, expressing complete uncertainty about a probability. Confirm it is equivalent to the uniform distribution.

Set alpha = 1, beta = 1. B(1, 1) = Gamma(1)*Gamma(1)/Gamma(2) = 1*1/1 = 1. The PDF becomes x^0 * (1-x)^0 / 1 = 1 for all x in [0, 1].

For any x in [0, 1], P(X ≤ x) = x. At x = 0.5, the CDF is exactly 50%. Mean = 1/2 = 0.5. Variance = 1/(4*3) = 1/12 ≈ 0.0833.

The mode is undefined since alpha = beta = 1 (every point in [0, 1] is equally dense). Skewness = 0. Excess kurtosis = -6/5 = -1.2, identical to the continuous uniform distribution.

PDF = 1.0 everywhere, P(X ≤ 0.5) = 50%, Mean = 0.5

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Example 3 - Symmetric Bell: Beta(3, 3)

An A/B testing model assumes a conversion rate follows Beta(3, 3). What is the probability the rate falls below 50%? Find the mean, mode, and standard deviation.

Set alpha = 3, beta = 3. Since alpha = beta, the distribution is symmetric about 0.5. Mean = 3/6 = 0.5. Mode = (3-1)/(6-2) = 2/4 = 0.5.

By symmetry, P(X ≤ 0.5) = exactly 50%. PDF at x = 0.5: f(0.5) = 0.5^2 × 0.5^2 / B(3,3). B(3,3) = 4/120 = 1/30. f(0.5) = 0.25 × 0.25 × 30 = 1.875.

Variance = 9/(36*7) = 9/252 = 1/28 ≈ 0.03571. SD = 1/√28 ≈ 0.1890. Skewness = 0 (symmetric). Median = 0.5 = mean = mode.

P(X ≤ 0.5) = 50%, Mean = Mode = Median = 0.5, SD ≈ 0.1890

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❓ Frequently Asked Questions

What is the beta distribution?+

The beta distribution is a continuous probability distribution defined on the interval [0, 1], parameterized by two positive shape parameters alpha and beta. It assigns flexible probability density to proportions and probabilities. Depending on alpha and beta, it can be uniform, bell-shaped, skewed left, skewed right, or U-shaped. It is widely used in Bayesian statistics, project management, and any model where outcomes are bounded between 0 and 1.

What do the shape parameters alpha and beta control?+

Alpha concentrates density near 1 and beta concentrates density near 0. When both are greater than 1, the distribution has a single interior peak. When alpha is larger than beta, the peak shifts toward 1 and the distribution is left-skewed. When beta is larger than alpha, the peak shifts toward 0 and the distribution is right-skewed. When alpha = beta, the distribution is symmetric about 0.5.

How do I compute the beta distribution PDF?+

The PDF is f(x; alpha, beta) = x^(alpha-1) * (1-x)^(beta-1) / B(alpha, beta), where B is the beta function equal to Gamma(alpha)*Gamma(beta)/Gamma(alpha+beta). For Beta(2, 5) at x = 0.3: f(0.3) = 0.3 * 0.2401 * 30 = 2.161. Note the PDF is a density, not a probability. Values above 1 are valid for concentrated distributions.

What is the CDF of the beta distribution?+

The CDF P(X at most x) equals the regularized incomplete beta function I_x(alpha, beta), which cannot be expressed in closed form for general alpha and beta. This calculator uses Lentz's continued fraction algorithm (identical to Numerical Recipes betai) for numerically stable evaluation. For Beta(2, 5) at x = 0.3, the CDF is 57.98%.

What is the mean of Beta(alpha, beta)?+

The mean is alpha / (alpha + beta). For Beta(2, 5), mean = 2/7 = 0.2857. For Beta(5, 2), mean = 5/7 = 0.7143. For Beta(1, 1), mean = 0.5. The mean is always between 0 and 1, and it represents where the distribution is centered on average.

What is the variance and standard deviation of the beta distribution?+

The variance is alpha*beta / ((alpha+beta)^2 * (alpha+beta+1)). For Beta(2, 5), variance = 10/392 = 0.02551 and SD = 0.1597. As the total alpha+beta grows (even at fixed mean), the variance decreases and the distribution concentrates. Doubling both alpha and beta halves the variance and narrows the spread.

How do I find the mode of the beta distribution?+

For alpha greater than 1 and beta greater than 1, the mode is (alpha-1)/(alpha+beta-2). For Beta(2, 5), mode = 1/5 = 0.2. When alpha is at most 1 and beta exceeds 1, the mode is at x = 0. When alpha exceeds 1 and beta is at most 1, the mode is at x = 1. When both parameters are at most 1, the distribution is U-shaped and has no interior mode.

What is the skewness of the beta distribution?+

Skewness = 2*(beta - alpha) * sqrt(alpha+beta+1) / ((alpha+beta+2) * sqrt(alpha*beta)). When alpha equals beta, skewness is zero (symmetric). When alpha is less than beta, skewness is positive (right tail is heavier). When alpha exceeds beta, skewness is negative (left tail is heavier).

How is the beta distribution used in Bayesian statistics?+

The beta distribution is the conjugate prior for the binomial likelihood. Starting with a Beta(alpha, beta) prior on a success probability p, after observing s successes and f failures the posterior is Beta(alpha+s, beta+f). This analytical update avoids numerical integration entirely. Starting with Beta(1, 1) (a uniform prior), after 8 successes and 2 failures the posterior is Beta(9, 3) with mean 9/12 = 0.75.

What is the Beta(1, 1) distribution?+

Beta(1, 1) is exactly the Uniform(0, 1) distribution. The PDF is constant at 1 over [0, 1], the CDF is simply x, and every interval of the same length has the same probability. It is often used as a non-informative prior in Bayesian analysis when all values of p in [0, 1] are considered equally plausible before seeing data.

What does it mean when alpha or beta is less than 1?+

When alpha is less than 1, the PDF rises steeply near x = 0 and then falls, concentrating probability mass near 0. When beta is less than 1, it concentrates mass near 1. When both are less than 1, the distribution is U-shaped: mass is concentrated near both endpoints 0 and 1, with a minimum in the interior. This models bimodal data where extreme values near the boundaries are more likely than middle values.

How does the beta distribution relate to the binomial distribution?+

The CDF of a Binomial(n, p) evaluated at k is related to the beta CDF: P(Binomial(n,p) at most k) = I_{1-p}(n-k, k+1). The beta and binomial distributions are connected through the incomplete beta function. In Bayesian terms, the beta distribution on p is the conjugate prior for the binomial likelihood on k given p, which is why their update rule is so clean.

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📌 Quick Tips

💡Both α and β must be strictly greater than zero. Large values concentrate the distribution near the mean; values close to zero spread it toward the endpoints 0 and 1.

💡When α equals β, the distribution is symmetric about 0.5. When α exceeds β, it leans toward 1. When β exceeds α, it leans toward 0.

💡For Bayesian inference of a binomial proportion, set α to prior successes plus 1 and β to prior failures plus 1 after observing data.

💡The special case Beta(1, 1) is identical to the Uniform(0, 1) distribution. Every x in [0, 1] is equally likely and the PDF is a flat line at 1.

💡The Distribution Stats mode is useful when you need median and skewness quickly without entering a specific x value.