Library / Probability distributions / PRD-25
Probability distributions
Beta proportion: probability density
Probability density of a fraction x between 0 and 1 under a beta distribution with shapes a and b.
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- PRD-25
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- v1
- Field
- Probability distributions
- Result
- f
The beta density describes a proportion between 0 and 1. The shapes a and b control the form: a = b = 1 is flat, a, b > 1 gives a single hump, and a or b below 1 gives a density that rises without limit at an end. B(a, b) is the beta function that makes the area equal one. The density is dimensionless; it can exceed 1.
Probability density of a fraction x between 0 and 1 under a beta distribution with shapes a and b. B(a, b) is the beta function that makes the area equal one.
Calculator
- Mean, a/(a + b)0.2857
- Mode, (a − 1)/(a + b − 2)0.2
- 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.
- At x = 0.3 the density is 2.161, against a mean proportion of 0.286.
- A density above 1 is not a mistake: probability is the area, and the range is only 1 wide.
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 | 2.97e-7 | 2.456 | 2.456 | +10% | −7.68 % |
| a Shape a | 0.5 to 10 | 0.04731 | 2.316 | 2.268 | +10% | +4.07 % |
| b Shape b | 0.5 to 10 | 0.2689 | 2.162 | 1.893 | +10% | −0.298 % |
Sweep of Proportion x
| Proportion x | Probability density |
|---|---|
| 0.01 | 0.2882 |
| 0.1325 | 2.251 |
| 0.255 | 2.357 |
| 0.3775 | 1.701 |
| 0.5 | 0.9375 |
| 0.6225 | 0.3793 |
| 0.745 | 0.0945 |
| 0.8675 | 0.008021 |
| 0.99 | 2.97e-7 |
Sweep one variable
Variables and how each one contributes
Probability density
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 +4.07 %, from 2.161 to 2.249.
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 −0.298 %, from 2.161 to 2.154.
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 −13.3 %, from 2.161 to 1.874.
Worked example with the values above
- Powers: x^(a−1) = 0.3 and (1 − x)^(b−1) = 0.2401.
- Beta function: B(2, 5) = 0.033333.
- f = (product) / B = 2.1609.
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