Library / Probability distributions / PRD-31
Probability distributions
Laplace error: lower tail probability
Probability that a Laplace-distributed reading is at or below x.
- Ref
- PRD-31
- Rev
- v1
- Field
- Probability distributions
- Result
- F
F is the probability that a reading is at or below x. It is ½e^((x − m)/s) at or below the location, and 1 − ½e^(−(x − m)/s) above it, equal to one half at x = m. Use it for a one-sided threshold.
Probability that a Laplace-distributed reading is at or below x. It equals one half at the location m.
Calculator
- Probability above x, 1 − F0.2567
- Standardised distance, (x − m)/s0.6667
- Median (equal to 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.
- About 74.3 % of readings are at or below 4 mm, and 25.7 % are above.
- The curve is symmetric about the location, so the probability below m − d equals the probability above m + d.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Reading x | −5 to 12 mm | 0.04849 | 0.9822 | 0.9337 | +0.34 mm | +3.7 % |
| m Location m | −5 to 10 mm | 0.06767 | 0.9751 | 0.9074 | +0.3 mm | −3.63 % |
| s Scale s | 0.5 to 10 mm | 0.5906 | 0.9908 | 0.4002 | +10% | −2.16 % |
Sweep of Reading x
| Reading x (mm) | Probability at or below x |
|---|---|
| −5 | 0.04849 |
| −2.875 | 0.09846 |
| −0.75 | 0.1999 |
| 1.375 | 0.406 |
| 3.5 | 0.6967 |
| 5.625 | 0.8507 |
| 7.75 | 0.9265 |
| 9.875 | 0.9638 |
| 12 | 0.9822 |
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 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 −2.38 %, from 0.7433 to 0.7256.
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 −2.16 %, from 0.7433 to 0.7273.
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 +4.31 %, from 0.7433 to 0.7753.
Worked example with the values above
- Standardised distance: (x − m)/s = (4 − 2)/3 = 0.66667.
- Reading is above m: F = 1 − ½e^(−0.66667).
- F = 0.74329.
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