Library / Probability distributions / PRD-26
Probability distributions
Beta proportion: probability below a target
Probability that a beta-distributed proportion is at or below a target x.
- Ref
- PRD-26
- Rev
- v1
- Field
- Probability distributions
- Result
- F
F is the regularised incomplete beta function Ix(a, b): the probability that the proportion is at or below x. It is 0 at x = 0 and 1 at x = 1. For integer shapes it also equals a binomial tail probability, which is how the same function turns up in acceptance testing.
Probability that a beta-distributed proportion is at or below a target x. I is the regularised incomplete beta function; it is 0 at x = 0 and 1 at x = 1.
Calculator
- Probability above the target, 1 − F0.4202
- Mean proportion0.2857
- Standard deviation0.1597
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.
- There is a 58 % probability that the proportion is at or below 0.3, and 42 % that it is above.
- The mean is 0.286, so the target of 0.3 is above the average proportion.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Proportion x | 0.01 to 0.99 | 0.00146 | 1 | 0.9985 | +10% | +10.8 % |
| a Shape a | 0.5 to 10 | 0.001666 | 0.9347 | 0.9331 | +10% | −8.58 % |
| b Shape b | 0.5 to 10 | 0.03784 | 0.887 | 0.8492 | +10% | +8.2 % |
Sweep of Proportion x
| Proportion x | Probability at or below x |
|---|---|
| 0.01 | 0.00146 |
| 0.1325 | 0.1832 |
| 0.255 | 0.4779 |
| 0.3775 | 0.7301 |
| 0.5 | 0.8906 |
| 0.6225 | 0.9685 |
| 0.745 | 0.9949 |
| 0.8675 | 0.9998 |
| 0.99 | 1 |
Sweep one variable
Variables and how each one contributes
Probability at or below x
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 −8.58 %, from 0.5798 to 0.5301.
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 +8.2 %, from 0.5798 to 0.6274.
Proportion x
The proportion at which the result is read, strictly between 0 and 1. Moves the point on the 0 to 1 scale. Raising it by 0.05 (to 0.35) moves the result by +17.4 %, from 0.5798 to 0.6809.
Worked example with the values above
- Shapes a = 2, b = 5, target x = 0.3.
- Evaluate the regularised incomplete beta function I(0.3; 2, 5).
- F = 0.57983.
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
The inputs are parameters, usually fitted to data, and their estimation error carries into the result. The probability or density is a smooth function of the parameters, so the sweep chart shows how sensitive it is: where the curve is steep, a small error in a parameter makes a large difference. The formula treats the parameters as exact, so it gives no confidence interval on its own.
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