Library / Probability distributions / PRD-23
Probability distributions
Gamma waiting time: variance
Variance of a gamma waiting time, in squared time units.
- Ref
- PRD-23
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [h²]
The variance ks² is in squared time units. The ratio of the variance to the mean is the scale s, so a large scale means a large spread relative to the average. The standard deviation s√k grows only as the square root of the number of events.
Variance of a gamma waiting time, in squared time units. The ratio of the variance to the mean is the scale s.
Calculator
- Standard deviation3.464 h
- Mean waiting time6 h
- Coefficient of variation, 1/√k0.5774
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.
- The standard deviation of the wait is 3.464 h against a mean of 6 h.
- Relative to the mean, the spread shrinks as 1/√k: waiting for more events makes the total time more predictable.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (h²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 h | 0.75 | 300 | 299.3 | +10% | +21 % |
| k Shape k | 0.5 to 10 | 2 | 40 | 38 | +10% | +10 % |
Sweep of Shape k
| Shape k | Variance (h²) |
|---|---|
| 0.5 | 2 |
| 1.688 | 6.75 |
| 2.875 | 11.5 |
| 4.063 | 16.25 |
| 5.25 | 21 |
| 6.438 | 25.75 |
| 7.625 | 30.5 |
| 8.813 | 35.25 |
| 10 | 40 |
Sweep one variable
Variables and how each one contributes
Variance
What the equation gives you. Everything else feeds it.
Shape k
The shape parameter (a plain number). For a whole number k it counts the events being waited for; below 1 the density is infinite at zero. Raises the mean and variance in proportion and makes the shape more symmetric. Raising it by 10 % (to 3.3) moves the result by +10 %, from 12 to 13.2 h².
Scale s
The scale, in hours: the mean waiting time per event. It is the reciprocal of the event rate. Stretches every time in the model in proportion. Raising it by 10 % (to 2.2) moves the result by +21 %, from 12 to 14.52 h².
Worked example with the values above
- Square the scale: s² = 4 h².
- Multiply by the shape: 3 × 4.
- Var = 12 h².
Assumptions and limits
- Waiting times follow a gamma distribution; for whole-number k this is the time to the kth event of a steady, memoryless (Poisson) process.
- The shape and scale are known and use the same time unit as x.
Common mistakes
- Confusing the scale s with the rate λ = 1/s: using a rate where a scale is expected inverts the mean.
- Using the gamma form with non-constant event rates, where the Poisson-process link no longer holds.
- Reading the density as a probability.
Statistics and uncertainty
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
Where it is used
- Time until the kth failure, arrival or repair in a steady random process.
- Rainfall totals, queue service times and other positive, right-skewed quantities.
Maths you need
- Gamma function: the generalised factorial
- Poisson process: events at a constant average rate