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Help screen describing the interactive elements #### Proof Close [Probability Playground:](https://www.acsu.buffalo.edu/~adamcunn/probability/probability.html) # The Beta Distribution - [Map](https://www.acsu.buffalo.edu/~adamcunn/probability/probability.html) - [About](https://www.acsu.buffalo.edu/~adamcunn/probability/about.html) - [Help](https://www.acsu.buffalo.edu/~adamcunn/probability/beta.html) Beta(α, β) Distribution [Bernoulli(p) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/bernoulli.html) [Beta(α, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/beta.html) [Beta-Binomial(n, α, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/betabinomial.html) [Binomial(n, p) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/binomial.html) [Cauchy(θ, σ) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/cauchy.html) [Chi-Squared(ν) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/chisquared.html) [Discrete Uniform(N₀, N₁) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/discreteuniform.html) [Exponential(β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/exponential.html) [F(ν₁, ν₂) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/f.html) [Fréchet(α, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/frechet.html) [Gamma(α, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/gamma.html) [Geometric(p) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/geometric.html) [Gumbel(μ, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/gumbel.html) [Hypergeometric(N, M, K) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/hypergeometric.html) [Laplace(μ, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/laplace.html) [Logistic(μ, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/logistic.html) [Lognormal(μ, σ²) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/lognormal.html) [Negative Binomial(r, p) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/negativebinomial.html) [Normal(μ, σ²) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/normal.html) [Pareto(α, xₘ) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/pareto.html) [Poisson(λ) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/poisson.html) [Standard Cauchy Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/standardcauchy.html) [Standard Logistic Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/standardlogistic.html) [Standard Normal Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/standardnormal.html) [Standard Pareto Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/standardpareto.html) [Standard Uniform Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/standarduniform.html) [T(ν) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/t.html) [Uniform(a, b) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/uniform.html) [Weibull(α, β) Distribution](https://www.acsu.buffalo.edu/~adamcunn/probability/weibull.html) The beta distribution is a continuous distribution on the interval \[0, 1\]. It arises naturally as the distribution of the order statistics of a [standard uniform](https://www.acsu.buffalo.edu/~adamcunn/probability/standarduniform.html) random sample. If *n* points are randomly chosen from the interval \[0, 1\] and arranged in order, then the *j*th point has a beta(*j*, *n* − *j* + 1) distribution. The function B(a, b) in the pdf and cdf denotes the [beta function](https://www.acsu.buffalo.edu/~adamcunn/probability/beta.html), while the function B(x; a, b) in the cdf denotes the [incomplete beta function](https://www.acsu.buffalo.edu/~adamcunn/probability/beta.html). | Parameter | Range | Description | |---|---|---| | α | α \> 0 | Shape parameter | | β | β \> 0 | Shape parameter | Probability Density Function f ( x ; α , β ) \= 1 B ( α , β ) x α − 1 ( 1 − x ) β − 1 F ( x ; α , β ) \= B ( x ; α , β ) B ( α , β ) Support 0 ≤ x ≤ 1 Mean α α \+ β Variance α β ( α \+ β ) 2 ( α \+ β \+ 1 ) | Example | α | β | |---|---|---| | Five numbers are chosen at random from the interval (0, 1) and arranged in order. The middle number has a beta(3, 3) distribution. | 3\.000 | 3\.000 | | Relative humidity for Auckland, New Zealand for the first six months of 2022 has an approximate beta(5.93, 1.78) distribution. | 5\.930 | 1\.780 | | The probability that an apple will be unblemished depends on yearly weather and insect populations, and has a beta(8, 0.5) distribution. | 8\.000 | 0\.5000 | Pdf Cdf Visualize Simulate ## X ∼ Beta(α, β) α = β =  Chart area for displaying the beta pdf, cdf, and simulation 0\.0 0\.1 0\.2 0\.3 0\.4 0\.5 0\.6 0\.7 0\.8 0\.9 1\.0 x 0\.0 0\.1 0\.2 0\.3 0\.4 0\.5 0\.6 0\.7 0\.8 0\.9 1\.0 1\.1 f(x) α β E(X) = 0\.5000, Var(X) = 0\.08333 Auto x-axis Fixed x-axis Auto y-axis Fixed y-axis ⏮ Reset ⏯ Step ▶ Run ⏩ Fast Because it is bounded between 0 and 1, this distribution is often used to model quantities representing fractions or percentages. The beta distribution is also used to model the distribution of probabilities, when the probability of an event is itself a random variable. The pdf can take many shapes, either horizontal, strictly increasing or decreasing, u-shaped, or unimodal. Because it is bounded between 0 and 1, this distribution is often used to model quantities representing fractions or percentages. The beta distribution is also used to model the distribution of probabilities, when the probability of an event is itself a random variable. The illustration above shows a sample of n points chosen randomly and independently from the interval \[0, 1\]. The *order statistics* X(1), …, X(n) of this sample are the sample values placed in ascending order. The jth order statistic X(j) has a beta(j, n − j + 1) distribution. The simulation above shows a set of *n* points chosen randomly from the interval \[0, 1\], where *n* = α + β − 1 after rounding α and β to the nearest positive integer. When the points are arranged in order, the location of point α has a beta(α, β) distribution. The histogram accumulates the results of each simulation. Transformation: Beta(β, α) Transformation: Beta(β, α) Limiting distribution: Normal(E(X), Var(X)) Special case: Standard Uniform ## Y = 1 − X ∼ Beta(β, α) limα,β→∞ X ∼ Normal(E(X), Var(X)) Beta(1, 1) ∼ Standard Uniform α = 1\.000, β = 1\.000 ![Visualization of the related distribution]() Chart area for displaying the related pdf, cdf, and simulation 0\.0 0\.1 0\.2 0\.3 0\.4 0\.5 0\.6 0\.7 0\.8 0\.9 1\.0 y 0\.0 0\.1 0\.2 0\.3 0\.4 0\.5 0\.6 0\.7 0\.8 0\.9 1\.0 1\.1 f(y) E(Y) = 0\.5000, Var(Y) = 0\.08333 Auto x-axis Fixed x-axis Auto y-axis Fixed y-axis If *X* is a beta(α, β) random variable, then *Y* = 1 − *X* is a beta(β, α) random variable. The beta(α, β) distribution converges to the normal distribution with mean α/(α + β) and variance αβ/\[(α + β)²(α + β + 1)\] as α, β → ∞. The beta(1, 1) distribution is a standard uniform distribution. This can be seen in the graph above, where swapping α and β gives a mirror image of the original beta distribution *X*. The red dashed line shows the distribution of Beta(α, β), which can be seen to converge to a [normal](https://www.acsu.buffalo.edu/~adamcunn/probability/normal.html) distribution as α, β → ∞. This convergence can be seen by reducing the variance in the graph of the beta distribution. The probability mass then becomes more concentrated around the mean, and the distribution becomes more bell-shaped. The beta(α, β) distribution is shown in red superimposed over the [standard uniform](https://www.acsu.buffalo.edu/~adamcunn/probability/standarduniform.html). It can be seen that these are identical when α = β = 1. Note that the *j*th point *X*(j) in an ordered standard uniform random sample of size *n* has a beta(*j, n − j + 1*) distribution. A beta(1, 1) distribution therefore has *j* = *n* = 1, which is the distribution of a standard uniform random variable. Copyright © 2018 - 2026, Adam Cunningham, University at Buffalo. - [](http://www.acsu.buffalo.edu/~adamcunn/) - [](https://www.linkedin.com/in/adam-cunningham-227b74174/) - [](mailto:adamcunn@buffalo.edu)