Question

Relation between gamma distribution and exponential distribution

Answer 1

The easiest way to understand the Gamma distribution is probably as a model for waiting time

Suppose you go down to the high way bridge and record how long time you have to wait before the first car drives by. This has an exponential distribution (red curve): the most likely waiting time is zero because if the chance that a car comes in any second is (say) 10%, then the chance that the first car comes during the first second is 10%, while the chance that the first car comes during (say) the 5th second is smaller because it requires not only that a car actually came during that second, but also that no other car came before.

But if you instead record how long time it takes before (say) two cars have come, then the most likely waiting time is not longer 0 because it is unlikely that two cars both come during the first second. You will get something like the blue curve. Similarly, you could record the waiting time to the 3rd, 4th or 5th car.

This is the Erlang distribution. The Gamma distribution generalizes this by also allowing non-integer number of cars. You could think of a Gamma distributed variable as for example “how long does it take for a bucket to fill up if I leave it out in the rain”.

Answer 2

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solution:--

A negative binomial distribution with n = 1 is a geometric distribution. Agamma distribution with shape parameter α = 1 and scale parameter β is an exponential (β) distribution. Agamma (α, β) random variable with α = ν/2 and β = 2, is a chi-squared random variable with ν degrees of freedom.