Library / Probability distributions / PRD-24
Probability distributions
Gamma waiting time: skewness
Dimensionless measure of how lopsided a gamma distribution is.
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- PRD-24
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- v1
- Field
- Probability distributions
- Result
- Skew
Skewness measures how lopsided the distribution is. For a gamma distribution it is 2/√k, depending only on the shape. A value of zero would be symmetric; positive means a long upper tail. It falls as k grows, so the distribution approaches a normal curve for large shape.
Dimensionless measure of how lopsided a gamma distribution is. It depends only on the shape k and shrinks as k grows, so the distribution approaches a symmetric bell.
Calculator
- Excess kurtosis, 6/k2
- Shape needed for skewness 0.516
- 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.
- A shape of 3 gives a skewness of 1.15, a clear upper tail.
- A shape of 16 or more brings the skewness down to 0.5 or less, where a normal approximation starts to be reasonable.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Shape k | 0.5 to 10 | 0.6325 | 2.828 | 2.196 | +10% | −4.65 % |
Sweep of Shape k
| Shape k | Skewness |
|---|---|
| 0.5 | 2.828 |
| 1.688 | 1.54 |
| 2.875 | 1.18 |
| 4.063 | 0.9923 |
| 5.25 | 0.8729 |
| 6.438 | 0.7883 |
| 7.625 | 0.7243 |
| 8.813 | 0.6737 |
| 10 | 0.6325 |
Sweep one variable
Variables and how each one contributes
Skewness
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 −4.65 %, from 1.155 to 1.101.
Worked example with the values above
- Take the square root of the shape: √3 = 1.7321.
- Divide 2 by it.
- Skew = 1.1547.
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