Beta Distribution Calculator

Compute beta distribution probabilities, density (pdf), cumulative probability (CDF), quantiles, and measures like mean, variance, mode, skewness, and kurtosis from the shape parameters alpha and beta.

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Mathematics

Statistics

Beta Distribution Calculator

Compute beta distribution probabilities, density (pdf), cumulative probability (CDF), quantiles, and measures like mean, variance, mode, skewness, and kurtosis from the shape parameters alpha and beta.

Beta Distribution Calculator

Beta distribution inputs

Density f(x) at x
P(X ≥ x)

Distribution plots

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The beta distribution is a family of continuous probability distributions defined on the interval between 0 and 1. It is determined by two positive shape parameters, α and β, which together govern its shape. In simple terms it can take many different shapes from flat curves to sharp bell curves or U-shaped curves. This calculator allows you to compute the density, cumulative probability, quantiles (inverse cumulative probability) as well as standardized summary measures.

What is a beta distribution?

Because a random variable following the beta distribution always lies between 0 and 1, it is well-suited to represent proportions, ratios or probabilities themselves. As the shape is determined by only two numbers, the same distribution can represent both symmetric and skewed shapes, unimodal or bimodal.

Changing the parameters α and β also changes the shape of the curve. When both parameters are equal, the curve is symmetric around the value of one half. If both parameters are greater than 1, a bell-shaped curve with only one peak in the interior results. On the other hand, setting both parameters to values less than 1 produces an U-shaped curve that dips at the middle and rises on either end.

The formula for density is:

The probability density function of the beta distribution is given by:

f(x)=xα−1 (1−x)β−1B(α,β)f(x) = \frac{x^{\alpha - 1}\,(1 - x)^{\beta - 1}}{B(\alpha, \beta)}

The numerator represents the shape. The denominator is the beta function, and the whole area is scaled to be equal to 1. The beta function can be expressed in terms of the gamma function as follows:

B(α,β)=Γ(α) Γ(β)Γ(α+β)B(\alpha, \beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha + \beta)}

The cumulative distribution function gives the probability that a variable is less than or equal to x and it is a regularized incomplete beta function. As there is no simple closed form for it, this tool performs numerical calculations and finds quantiles using the inverse operation.

F(x)=P(X≤x)=Ix(α,β)F(x) = P(X \le x) = I_x(\alpha, \beta)

How to use this calculator:

Select which value to calculate and enter the two shape parameters.

In the Probability mode you can select the type of probability and check the left probability P(X ≤ x), the right probability P(X ≥ x) or the probability P between two values. In Quantile mode, when you enter a cumulative probability p, it will return the value of x for which this probability holds. In Distribution measures mode you can obtain simultaneously mean, variance, standard deviation, mode, median, skewness and kurtosis. The bottom two graphs always show the current density function and the cumulative function.

Overview of the forms.

The table shows common parameter combinations and the resulting shapes. By swapping alpha and beta, the graph is mirror symmetric with respect to an axis.

alpha

beta

Shape

1

1

Flat (standard uniform)

0.5

0.5

U-shaped (arcsine)

2

2

Symmetric bell about 0.5

2

5

Right-skewed single peak

5

2

Left-skewed single peak

1

3

Strictly decreasing (J-shaped)

Summary of key figures:

The mean, or expected value, is at the position of density equilibrium.

μ=αα+β\mu = \frac{\alpha}{\alpha + \beta}

Variance gives the size of the spread of values.

σ2=α β(α+β)2 (α+β+1)\sigma^2 = \frac{\alpha\,\beta}{(\alpha + \beta)^2\,(\alpha + \beta + 1)}

If both parameters are greater than 1, then the mode, or position of the peak, is given by:

mode=α−1α+β−2\text{mode} = \frac{\alpha - 1}{\alpha + \beta - 2}

The tilt indicates the direction and degree of asymmetry.

γ1=2 (β−α) α+β+1(α+β+2) α β\gamma_1 = \frac{2\,(\beta - \alpha)\,\sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2)\,\sqrt{\alpha\,\beta}}

If the skewness is exactly 0, then the distribution is symmetric. A positive value indicates a longer tail to the right, while a negative value indicates a longer tail to the left. The median cannot usually be calculated by an simple formula, and must instead be estimated using the inverse cumulative distribution function.

Example calculation:

If alpha is equal to 2 and beta is equal to 3, calculate the probability that X is less than or equal to half of its maximum value. The mean is the result of dividing 2 by 5, which is 0.4. The variance is the result of dividing 1 by 25, which is 0.04, so the standard deviation is 0.2. The cumulative probability at 0.5 is 0.6875. This means that the probability that X is greater than half of its maximum value is 0.3125.

P(X≤0.5)=I0.5(2,3)=0.6875P(X \le 0.5) = I_{0.5}(2, 3) = 0.6875

Applications of beta distribution:

The beta distribution is often used in Bayesian statistics because it is the conjugate prior distribution for the likelihoods of binomial, negative binomial and geometric distributions. One starts with a beta prior distribution, then when some successes and failures are observed, the posterior distribution is also a beta distribution. Since only the number of successes or failures need to be added to the two parameters, the updating cost is low and the method is easy to understand.

Besides Bayes inference, the beta distribution can be used to model proportions and rates in various fields. Project planners use it in PERT scheduling to represent the time required to complete a task, while analysts use it for conversion rates, failure rates, and any quantity that needs to be modeled between 0 and 1.

Frequently asked questions

What is a beta distribution?

The beta distribution is a family of continuous probability distributions defined on the interval [0, 1]. This family of distributions can be symmetric, skewed, unimodal or bimodal and its shape is controlled by two positive parameters alpha and beta. Due to this flexibility it is often used for modeling proportions or probabilities.

How to calculate mean of beta distribution?

The mean or expected value is the result of dividing alpha by the sum of alpha and beta. So for example if alpha was 2 and beta was 3 then the mean would be the result of dividing 2 by 5 which is 0.4.

How do you determine if a beta distribution is symmetric or skewed?

The two parameters are compared. If alpha and beta are equal, then the distribution is symmetric with a skewness of zero. If alpha is less than beta, the distribution is skewed to the right, with the long tail towards 1. If alpha is greater than beta, the distribution is skewed to the left, with the long tail towards 0.

Why is the beta distribution used in Bayesian inference?

The beta distribution is a conjugate prior distribution for the likelihood function of the binomial distribution, negative binomial distribution and geometric distribution. If one starts with a beta prior distribution and observes data, then the posterior distribution will also be a beta distribution. Thus, only the number of successes and failures need to be added as two parameters, rather than computing the posterior distribution from scratch.

What is the difference between density and cumulative probability?

The density f(x) gives the height at a particular point of a curve and represents where values are concentrated. The cumulative probability F(x) is the area under the density curve up to x, so it's the probability that a variable is less than or equal to x. The quantile value is the opposite: for a given cumulative probability, it gives the corresponding value of x.

Related calculators

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Mean CalculatorFind the mean (average) of your data set, along with the median, mode, range, standard deviation, and more. Enter your values and read the full summary.
Sample Variance CalculatorFind the sample and population variance of your data set, along with the standard deviation, mean, and sum of squares. Enter your values and read the full breakdown.
Expected Value CalculatorFind the expected value of a discrete random variable from its outcomes and probabilities, with the variance, standard deviation, and a step-by-step breakdown. Enter probabilities or weights and read the full summary.
Average Deviation CalculatorFind the average deviation (mean absolute deviation) of your data set about the mean or the median, with the mean, median, range, a step-by-step breakdown, and related measures.
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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

  1. Wikipedia: Beta distribution

    Definitions, density, moments, and properties of the beta distribution.

  2. Wolfram MathWorld: Beta Distribution

    Formula reference for the density, moments, and related functions.

  3. NIST/SEMATECH e-Handbook of Statistical Methods

    Standard reference on distribution functions and shape parameters.