Probability Poisson Process and Gamma Distribution

[Saladsamurai]

Homework Statement

The Attempt at a Solution

Part (a) is no problem, it is simply P(X>10) = 1 - P(X<=10) which requires the use of tabulated cumulative poisson values.

Part (b) is throwing for a loop. I know that I need to invoke the Gamma distribution since that is what the solution is doing. But I don't really understand why. I think that it is because there is some relationship between a Poisson process and the Gamma distribution, but I am not exactly sure what it is.

What I do know, is that for Poisson processes, the probability that an event occurs X = x number of times in 't' time depends on the average number of times the Poisson event occurs per unit time. But how can I use this knowledge to solve part (b)?

Any thoughts?
Casey



*************************************
*************************************
Here is the solution if it helps to generate any ideas. I am just not sure why they are using the Gamma distribution.

 

[Saladsamurai]

Any thoughts on this one? For some reason, I feel like it would make more sense to use the exponential distribution, but again, I am not really sure why. I still don't see why the fact that it is a poisson process allows me to infer that I can or should be using a gamma distribution?