Library / Probability distributions / PRD-27
Probability distributions
Beta proportion: mean
Expected value of a beta-distributed proportion.
- Ref
- PRD-27
- Rev
- v1
- Field
- Probability distributions
- Result
- E[X]
The mean is a/(a + b): the share of the total shape that belongs to a. In a Bayesian update with a successes and b failures plus a flat prior it is the expected probability of success. Adding observations moves it and, because a + b grows, narrows the distribution.
Expected value of a beta-distributed proportion. It is the share of the total shape a + b that belongs to a.
Calculator
- Total shape, a + b7
- Standard deviation0.1597
- Mode, (a − 1)/(a + b − 2)0.2
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.
- Shapes 2 and 5 give an average proportion of 0.286, equivalent to 28.6 %.
- Scaling both shapes by the same factor keeps the mean but narrows the distribution: it represents more evidence.
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 | 0.1667 | 0.8 | 0.6333 | +10% | −6.67 % |
| a Shape a | 0.5 to 10 | 0.09091 | 0.6667 | 0.5758 | +10% | +6.94 % |
Sweep of Shape a
| Shape a | Mean proportion |
|---|---|
| 0.5 | 0.09091 |
| 1.688 | 0.2523 |
| 2.875 | 0.3651 |
| 4.063 | 0.4483 |
| 5.25 | 0.5122 |
| 6.438 | 0.5628 |
| 7.625 | 0.604 |
| 8.813 | 0.638 |
| 10 | 0.6667 |
Sweep one variable
Variables and how each one contributes
Mean 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 +6.94 %, from 0.2857 to 0.3056.
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 −6.67 %, from 0.2857 to 0.2667.
Worked example with the values above
- Total shape: a + b = 2 + 5 = 7.
- Divide a by it.
- E[X] = 0.28571.
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