Library / Probability distributions / PRD-34
Probability distributions
Logistic threshold: probability
Probability that a logistic threshold is at or below x.
- Ref
- PRD-34
- Rev
- v1
- Field
- Probability distributions
- Result
- F
The logistic distribution function is the S-shaped sigmoid 1/(1 + e^(−z)) with z = (x − m)/s. It gives 50 % at x = m and approaches 0 and 1 in the tails. It is the curve fitted in logistic regression, and a stable implementation avoids overflow of the exponential at extreme arguments.
Probability that a logistic threshold is at or below x. It follows an S-shaped curve centred on m, with 50 % at x = m.
Calculator
- Standardised value, z = (x − m)/s0.6667
- Odds of being below x, F/(1 − F)1.948
- Probability above x, 1 − F0.3392
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.
- A threshold of 4 MPa lies at 66.1 % on the S-curve centred at 2 MPa.
- The log-odds of being below x are exactly z = 0.667: that straight-line relationship is what logistic regression fits.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Threshold x | −5 to 12 MPa | 0.0884 | 0.9656 | 0.8772 | +0.34 MPa | +3.77 % |
| m Location m | −5 to 10 MPa | 0.1192 | 0.9526 | 0.8334 | +0.3 MPa | −3.44 % |
| s Scale s | 0.5 to 10 MPa | 0.5498 | 0.982 | 0.4322 | +10% | −2.08 % |
Sweep of Threshold x
| Threshold x (MPa) | Probability at or below x |
|---|---|
| −5 | 0.0884 |
| −2.875 | 0.1645 |
| −0.75 | 0.2856 |
| 1.375 | 0.4481 |
| 3.5 | 0.6225 |
| 5.625 | 0.77 |
| 7.75 | 0.8718 |
| 9.875 | 0.9325 |
| 12 | 0.9656 |
Sweep one variable
Variables and how each one contributes
Probability at or below x
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 −2.29 %, from 0.6608 to 0.6457.
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 −2.08 %, from 0.6608 to 0.647.
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 +4.42 %, from 0.6608 to 0.69.
Worked example with the values above
- Standardised value: z = (4 − 2)/3 = 0.66667.
- Exponential: e^(−z) = 0.51342.
- F = 1/(1 + 0.51342) = 0.66076.
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