The basic concept
The standard beta distribution describes a continuous variable between 0 and 1. Its two positive shape parameters are α (alpha) and β (beta). Changing them creates symmetric, skewed, flat or U-shaped densities.
This support makes beta models candidates for proportions, recovery fractions or uncertainty about an unknown probability. A beta-distributed proportion differs from a binomial count.
Shape and parameters
| Parameters | Typical shape |
|---|---|
| α = β = 1 | Uniform density |
| α = β > 1 | Symmetric, concentrated towards the middle |
| α < β, both > 1 | Peak towards the lower end and a longer right side |
| α > β, both > 1 | Peak towards the upper end and a longer left side |
| α < 1 and β < 1 | U-shaped, with endpoint density singularities |
B(α,β) is the beta function, the normalising constant that makes total density area equal to one. An infinite density near an endpoint does not imply a positive point probability there.
Mean and variance
For a fixed mean, increasing α+β concentrates the distribution. If α and β both exceed 1, the interior mode is (α−1)/(α+β−2). That mode formula does not describe the other parameter cases.
Worked example: a recovery proportion
Assume recovery R has a Beta(2,5) distribution. All figures are teaching assumptions.
| Measure | Result |
|---|---|
| Mean recovery | 2/7 = 28.57% |
| Variance | 10/(49 × 8) = 0.02551 |
| Standard deviation | 15.97 percentage points |
| P(recovery ≤ 40%) | 76.67% |
| Mean loss fraction, 1 − R | 71.43% |
For a defaulted exposure of ₹10 lakh with this conditional recovery model, expected loss is ₹10 lakh × (1 − 2/7) = ₹7.1429 lakh. This is conditional on default. An unconditional expected credit loss also needs a default probability and appropriate dependence assumptions.
Excel illustration
| Task | Formula |
|---|---|
| Cumulative probability at 40% | =BETA.DIST(0.4,2,5,TRUE) |
| Density at 40% | =BETA.DIST(0.4,2,5,FALSE) |
| 95th-percentile recovery | =BETA.INV(0.95,2,5) |
| Mean recovery | =2/(2+5) |
The cumulative distribution function (CDF) returns probability. The probability density function (PDF) returns height. For bounds a and b, define Y = a + (b−a)R and adjust the scale.
Learning about an unknown default probability
A separate application treats an unknown default probability p as uncertain. With a Beta(α,β) prior and k defaults in n conditionally independent Bernoulli trials sharing the same p, the posterior is Beta(α+k, β+n−k).
For a Beta(1,19) prior and 3 defaults in 100 loans, the posterior is Beta(4,116). Its mean is 4/120 = 3.33%. This probability parameter is not the recovery proportion used in the previous example.
Limitations and checks
Can beta model a proportion of exactly 0 or 1 with positive probability?
A standard continuous beta model assigns zero probability to either exact endpoint. If real data contain genuine endpoint masses, consider a model with explicit zero or one inflation.
Can the same beta model explain correlated borrower defaults automatically?
No. The interpretation and dependence structure must be specified. A shared random default probability creates a different predictive count model from a fixed-probability binomial.
Interactive distribution laboratory
Change the parameters and threshold, then switch between cumulative probability and density or mass. Values are numerical approximations for learning.
Horizontal axis: the variable’s value. Vertical axis: density, probability mass, or cumulative probability as labelled. Shading highlights values at or below the threshold. The plot shows a finite viewing window.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Distribution definition and properties
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.