Library / Probability distributions / PRD-29
Probability distributions
Beta proportion: most probable value
Value at which the beta density peaks, defined inside the range only when both shapes exceed 1.
- Ref
- PRD-29
- Rev
- v1
- Field
- Probability distributions
- Result
- Mode
The mode (a − 1)/(a + b − 2) is the proportion at which the density peaks. It exists inside the range only when both a and b exceed 1; if either is 1 or less, the density has no interior peak. In Bayesian terms it is the maximum a posteriori estimate.
Value at which the beta density peaks, defined inside the range only when both shapes exceed 1.
Calculator
- Mean proportion0.2857
- Mode minus mean−0.08571
- Total shape, a + b7
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The density peaks at 0.2, compared with a mean of 0.286: the mode is below the mean when the distribution is skewed.
- As the shapes grow large the mode and the mean converge on the same value.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Shape b | 0.5 to 10 | 0.1 | 2 | 1.9 | +10% | −9.09 % |
| a Shape a | 0.5 to 10 | −0.1429 | 0.6923 | 0.8352 | +10% | +15.4 % |
Sweep of Shape a
| Shape a | Most probable proportion |
|---|---|
| 0.5 | −0.1429 |
| 1.688 | 0.1467 |
| 2.875 | 0.3191 |
| 4.063 | 0.4336 |
| 5.25 | 0.5152 |
| 6.438 | 0.5762 |
| 7.625 | 0.6235 |
| 8.813 | 0.6614 |
| 10 | 0.6923 |
Sweep one variable
Variables and how each one contributes
Most probable proportion
What the equation gives you. Everything else feeds it.
Shape a
The first shape parameter (a plain number). Think of it as a count of successes plus one in a Bayesian update. Pulls the proportion towards 1 and tightens the distribution. Raising it by 10 % (to 2.2) moves the result by +15.4 %, from 0.2 to 0.2308.
Shape b
The second shape parameter (a plain number): a count of failures plus one in a Bayesian update. Pulls the proportion towards 0 and tightens the distribution. Raising it by 10 % (to 5.5) moves the result by −9.09 %, from 0.2 to 0.1818.
Worked example with the values above
- Numerator: a − 1 = 1.
- Denominator: a + b − 2 = 5.
- Mode = 0.2.
Assumptions and limits
- The quantity is a proportion between 0 and 1 described by a beta distribution with a, b > 0.
- The shapes are dimensionless and come from data or from a prior belief.
Common mistakes
- Using percentages (0 to 100) for x: it must be on the 0 to 1 scale.
- Applying the mode formula when a or b is 1 or less: the peak is then at an end of the range.
- Reading a density above 1 as an error: densities can exceed 1 on a narrow range.
Statistics and uncertainty
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
Where it is used
- Describing uncertainty in a yield, defect rate or failure fraction, and updating it as test results arrive.
- Prior and posterior distributions for a probability in Bayesian reliability work.
Maths you need
- Probability density: area under the curve gives probability
- Beta function: a ratio of gamma functions