Factsheet: Beta distribution
Where to use: The beta distribution is used to model the distribution of probabilities or proportions. Hence, the random variable \(0 \leq X \leq 1\).
Notation: \(X \sim \textrm{Beta}(\alpha,\beta)\)
Parameters: Two positive real numbers \(\alpha,\beta\), which are shape parameters. These can be specified as follows in terms of \(n\) and \(k\) where \(n\) is the number of Bernoulli trials and \(k\) is the number of successes:
- \(\alpha = k + 1\)
- \(\beta = n - k + 1\)
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = \dfrac{\alpha}{\alpha+\beta}\) | |
| Variance | \(\mathbb{V}(X) = \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\) | |
| \(\mathbb{P}(X=x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{\textrm{B}(\alpha,\beta)}\) | \(\textrm{B}(x,y)\) is the beta function | |
| CDF | \(\mathbb{P}(X \leq x)=I_{x}(\alpha,\beta)\) | \(I_{x}(a,b)\) is the regularized incomplete beta function |
Example: Cantor’s Confectionery is visited by 10 customers, and 6 of them purchase something from the store. Taking the buying customers as successes and the total visiting customers as number of trials, there would be 6 successes, allowing you to find the following parameters:
\(\alpha = 6 + 1 = 7\)
\(\beta = 10 - 6 + 1 = 5\)
Then the distribution of the probabilities of a customer purchasing from Cantor’s Confectionery can be expressed as \(X \sim \textrm{Beta}(7,5)\), meaning the first shape parameter is 7 and the second shape parameter is 5.
Further reading
Version history
v1.0: initial version created 04/25 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.
v1.1: moved to factsheet form and populated with material from Overview: Probability distributions by tdhc.
v1.2: graph transferred from R Shinylive to html by tdhc in 06/26.