I am specifically confused as to the meanings of the shaping and scaling parameters of the gamma distribution and how they are used in a real context. Once I find the correct parameters, I have the equations to find everything else I need.
The word problem that I am looking at is:
Audrey, an astronomer is searching for extra-solar planets using the technique of relativistic lensing. Though there are believed to be a very large number of planets that can be found this way, actually finding one takes time and luck; and finding one planet does not help at all with finding planets of other stars in the same part of the sky. Audrey is good at it, and finds one planet at a time, on average once every three months.
Find the expected value and standard deviation of the number of planets she will find in the next two years.
Through logic alone, one can figure out that the expected value is eight, which should be equal to alpha times beta, if I am understanding the gamma distribution correctly.
Our teacher taught us gamma distribution with replacement parts. For instance, a system with 4 spare parts (plus the one already in the system) where each part lasts on average 4 months would be represented by a gamma distribution with shaping parameter 5, and scale parameter 4.
On the other hand, I may have misinterpreted the problem and am trying to use the wrong distribution. That's just the distribution we have been using, so I assumed that's the one to use. Instead, I am thinking that this may be a Poisson distribution.
Then the second part of the problem is:
When she finds her sixth new planet, she will be eligible for a prize. Find the expected value and standard deviation of the amount of time until she is eligible for that prize.
I would think this would use the gamma distribution with alpha = 6 and Beta = 3.