Library / Probability distributions / PRD-35
Probability distributions
Logistic threshold: density
Probability density of the logistic distribution, written with its cumulative probability F.
- Ref
- PRD-35
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/MPa]
The logistic density is F(1 − F)/s, the slope of the S-curve. It peaks at x = m with a height of 1/(4s) and falls away symmetrically. It is a bell-shaped curve with slightly heavier tails than the normal.
Probability density of the logistic distribution, written with its cumulative probability F. It is the slope of the S-curve and peaks at x = m.
Calculator
- Peak density, 1/(4s)0.08333 1/MPa
- Cumulative probability F0.6608
- Standard deviation, πs/√35.441 MPa
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 4 MPa the density is 0.07472 per MPa, 89.7 % of its peak value of 0.08333.
- A larger scale flattens the curve and spreads the transition over a wider range of x.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 MPa | 0.02475 | 0.1108 | 0.08606 | +10% | −7.38 % |
| x Threshold x | −5 to 12 MPa | 0.01109 | 0.08324 | 0.07215 | +0.34 MPa | −3.85 % |
| m Location m | −5 to 10 MPa | 0.01506 | 0.08333 | 0.06827 | +0.3 MPa | +3.03 % |
Sweep of Threshold x
| Threshold x (MPa) | Probability density (1/MPa) |
|---|---|
| −5 | 0.02686 |
| −2.875 | 0.04582 |
| −0.75 | 0.06802 |
| 1.375 | 0.08244 |
| 3.5 | 0.07833 |
| 5.625 | 0.05903 |
| 7.75 | 0.03726 |
| 9.875 | 0.02099 |
| 12 | 0.01109 |
Sweep one variable
Variables and how each one contributes
Probability density
What the equation gives you. Everything else feeds it.
Location m
The location, in MPa: the threshold at which the probability is one half. Shifts the S-curve left or right. Raising it by 10 % (to 2.2) moves the result by +2.06 %, from 0.07472 to 0.07626 1/MPa.
Scale s
The scale, in MPa: it sets how gradual the transition is. About 4.4 scales separate the 10 % and 90 % points. Flattens the S-curve and widens the transition. Raising it by 10 % (to 3.3) moves the result by −7.38 %, from 0.07472 to 0.06921 1/MPa.
Threshold x
The threshold value, in MPa, at which the result is read. Moves the point on the S-curve. Raising it by 10 % (to 4.4) moves the result by −4.57 %, from 0.07472 to 0.0713 1/MPa.
Worked example with the values above
- Cumulative probability: F = 0.66076.
- Product F(1 − F) = 0.22416.
- f = that / s = 0.074719 per MPa.
Assumptions and limits
- The threshold follows a logistic distribution: symmetric, with slightly heavier tails than a normal curve.
- The location and scale come from a fit to pass-fail or threshold data.
Common mistakes
- Confusing the scale s with the standard deviation: the logistic standard deviation is πs/√3.
- Evaluating the sigmoid with a naive exponential at extreme arguments, which can overflow; use a stable form.
- Using the fitted curve far outside the range of the data.
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
- Dose-response and pass-fail curves, and the sigmoid of logistic regression.
- Strength and fatigue thresholds where the chance of failure rises smoothly with load.
Maths you need
- Exponential function: e raised to a power
- Sigmoid curve: an S-shaped transition from 0 to 1