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Beta Distribution Calculator

Adjust α (alpha) and β (beta) shape parameters to examine mean, variance, skewness, and density values of the Beta(α, β) distribution.

Summary Statistics

Mean: 0.2857

Variance: 0.025510

Std. Deviation: 0.1597

Skewness: 0.5963

Excess Kurtosis: -0.1200

Use the table below to inspect Beta(α, β) density values across the interval [0, 1]. The density scales up to 2.458 at its peak with current parameters.

xf(x)
0.000.0000
0.010.2882
0.020.5534
0.030.7968
0.041.0192
0.051.2218
0.061.4053
0.071.5709
0.081.7193
0.091.8515
0.101.9683
0.112.0705
0.122.1589
0.132.2343
0.142.2974
0.152.3490
0.162.3898
0.172.4204
0.182.4415
0.192.4537
0.202.4576
0.212.4539
0.222.4430
0.232.4256
0.242.4021
0.252.3730
0.262.3390
0.272.3003
0.282.2574
0.292.2108
0.302.1609
0.312.1080
0.322.0526
0.331.9950
0.341.9354
0.351.8743
0.361.8119
0.371.7486
0.381.6845
0.391.6200
0.401.5552
0.411.4904
0.421.4259
0.431.3617
0.441.2982
0.451.2353
0.461.1734
0.471.1126
0.481.0529
0.490.9945
0.500.9375
0.510.8820
0.520.8281
0.530.7759
0.540.7253
0.550.6766
0.560.6297
0.570.5846
0.580.5414
0.590.5002
0.600.4608
0.610.4234
0.620.3878
0.630.3542
0.640.3225
0.650.2926
0.660.2646
0.670.2384
0.680.2139
0.690.1912
0.700.1701
0.710.1507
0.720.1328
0.730.1164
0.740.1014
0.750.0879
0.760.0756
0.770.0646
0.780.0548
0.790.0461
0.800.0384
0.810.0317
0.820.0258
0.830.0208
0.840.0165
0.850.0129
0.860.0099
0.870.0075
0.880.0055
0.890.0039
0.900.0027
0.910.0018
0.920.0011
0.930.0007
0.940.0004
0.950.0002
0.960.0001
0.970.0000
0.980.0000
0.990.0000
1.000.0000

How to Use This Calculator

  1. Provide positive shape parameters α and β to define the Beta(α, β) distribution.
  2. Review summary statistics to understand central tendency and spread.
  3. Use the density table to evaluate likelihood at specific x values between 0 and 1.
  4. Experiment with different shape parameters to visualize uniform, U-shaped, or skewed beta distributions.

Formula

f(x; α, β) = (Γ(α + β) / (Γ(α) Γ(β))) xα − 1 (1 − x)β − 1

Mean = α / (α + β)

Variance = (αβ) / ((α + β)2 (α + β + 1))

The beta distribution is defined on [0, 1] and acts as a conjugate prior for Bernoulli/binomial likelihoods, making it invaluable for Bayesian modeling.

Frequently Asked Questions

What happens if α = β?

The distribution is symmetric around 0.5. When α = β = 1, it reduces to the uniform distribution on [0, 1].

Can α or β be less than 1?

Yes. Parameters between 0 and 1 create U-shaped distributions, assigning more mass near 0 and 1 with thin middles.

How does this relate to binomial data?

Beta distributions are conjugate priors for binomial proportions. Update α and β with observed successes and failures to obtain posterior distributions.

Where is the mode?

Mode = (α − 1) / (α + β − 2) for α, β > 1. The calculator focuses on mean and variance, but you can compute the mode separately if needed.