The gamma distribution is a "waiting time" distribution. Suppose events occur independently and randomly with an average time between events of β. The waiting time until α events have occurred is a gamma(α, β) random variable.
The parameter α is known as the shape parameter, and the parameter β is called the scale parameter. Increasing α leads to a more "peaked" distribution, while increasing β increases the "spread" of the distribution.
| Parameter | Range | Description |
|---|---|---|
| α | α > 0 | Shape parameter |
| β | β > 0 | Scale parameter |
Probability Density Function
Support
Mean
Variance
| Example | α | β |
|---|---|---|
| A radioactive substance emits two alpha particles every second on average. Let X be the waiting time for three particles to be emitted. | 3.000 | 0.5000 |
| Cars arrive at an intersection at an average rate of one every two minutes. Let X be the waiting time until five cars have arrived. | 5.000 | 2.000 |
| Garage door lightbulbs last five years on average and are replaced when they fail. Let X be the time that a box of six bulbs lasts. | 6.000 | 5.000 |
X ~ Gamma(α, β)
E(X) = , Var(X) =
Note that the mean αβ is directly proportional to both α and β. This is what we would intuitively expect - the mean time spent waiting for α events to occur increases in proportion to both the number of events α and the average time β between events.
The illustration above shows events occuring randomly in time independently of other events, with a mean time between events of β. The time at which event number α occurs has a gamma(α, β) distribution.
The simulation above shows events occuring randomly in time independently of other events, with a mean time between events of β. The light blue line shows the time X at which event α occurs, with α rounded to the nearest positive integer. X then has a gamma(α, β) distribution. The histogram accumulates the results of each simulation.