Library / Probability distributions / PRD-30
Probability distributions
Laplace error: probability density
Probability density of a symmetric error with exponential tails on both sides of a location m, with scale s.
- Ref
- PRD-30
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/mm]
The Laplace density is a two-sided exponential spike centred on the location m. It falls as e^(−|x − m|/s), so it has a sharp peak and heavier tails than a normal curve with the same variance. The density has units of 1/x.
Probability density of a symmetric error with exponential tails on both sides of a location m, with scale s.
Calculator
- Peak density, 1/(2s)0.1667 1/mm
- Standard deviation, s√24.243 mm
- Distance of the reading from the location, |x − m|2 mm
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 mm the density is 0.08557 per mm, against a peak of 0.1667 per mm at the location.
- The density halves every 2.08 mm from the centre, which is the exponential tail at work.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Location m | −5 to 10 mm | 0.008298 | 0.1667 | 0.1584 | +0.3 mm | +10.5 % |
| x Reading x | −5 to 12 mm | 0.005946 | 0.1559 | 0.15 | +0.34 mm | −10.7 % |
| s Scale s | 0.5 to 10 mm | 0.01832 | 0.09189 | 0.07358 | +10% | −3.41 % |
Sweep of Reading x
| Reading x (mm) | Probability density (1/mm) |
|---|---|
| −5 | 0.01616 |
| −2.875 | 0.03282 |
| −0.75 | 0.06664 |
| 1.375 | 0.1353 |
| 3.5 | 0.1011 |
| 5.625 | 0.04978 |
| 7.75 | 0.02452 |
| 9.875 | 0.01207 |
| 12 | 0.005946 |
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 mm: the centre of the errors and their median. Shifts the whole distribution without changing its shape. Raising it by 10 % (to 2.2) moves the result by +6.89 %, from 0.08557 to 0.09147 1/mm.
Scale s
The scale, in mm. The mean absolute error equals s. Widens the distribution and lowers the peak. Raising it by 10 % (to 3.3) moves the result by −3.41 %, from 0.08557 to 0.08265 1/mm.
Reading x
The reading, in mm, at which the result is read. Moves the point at which the density or probability is read. Raising it by 10 % (to 4.4) moves the result by −12.5 %, from 0.08557 to 0.07489 1/mm.
Worked example with the values above
- Distance from the location: |x − m| = |4 − 2| = 2 mm.
- Exponential factor: e^(−2/3) = 0.51342.
- f = that / (2 × 3) = 0.08557 per mm.
Assumptions and limits
- Errors are independent, symmetric about m and have exponential tails.
- The scale is positive and comes from data for the same measurement process.
Common mistakes
- Using a normal model when outliers are more common than it predicts, which understates the tails.
- Mixing up the scale s with the standard deviation: the Laplace standard deviation is s√2.
- Reading the density at a point as a probability.
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
- Measurement errors and residuals with more large excursions than a normal curve gives.
- The noise model behind least-absolute-deviation (L1) fitting.
Maths you need
- Absolute value: the size of a number, ignoring its sign
- Exponential function: e raised to a power