# How does one get the gamma distribution parameters from a word problem description?

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.

Study this first: http://en.wikipedia.org/wiki/Poisson_process

From "finding one planet does not help at all with finding planets of other stars in the same part of the sky," you can model $N(t)$, the number of planets found up to time $t$, as a Poisson process. So, $N(t)$ has distribution $\text{Poisson}(\lambda t)$. I suggest you to measure time in months.

From "Audrey is good at it, and finds one planet at a time, on average once every three months," you can figure out the value of $\lambda$, and from the properties of the $\textrm{Poisson}(\lambda \cdot 24)$ distribution you can "find the expected value and standard deviation of the number of planets she will find in the next two years."

As you've studied in the Wikipedia page, for the Poisson process the intervals between events (finding a new planet) have independent exponential distributions with parameter $\lambda$. The sum of $k$ of these independent exponentials has distribution $\textrm{Gamma}(k,\lambda)$, which gives you "the expected value and standard deviation of the amount of time until she is eligible for that prize."

• While Zen's answer does not refer to Wikipedia's page on Gamma random variables, it is worth keeping in mind that Wikipedia would denote the distribution referred to above as $\Gamma(k,\lambda^{-1})$ Mar 27 '12 at 17:08
• Dilip is 100% right. The fact that the gamma distribution has two common parameterizations is a source of confusion. I'm using the definition (Schervish, "Theory of Statistics"): $Y\sim\textrm{Gamma}(a,b)$ iff it has density $f_Y(y)=\frac{b^a}{\Gamma(a)}y^{a-1} e^{-by} I_{[0,\infty)}(y)$. So, $Z\sim \textrm{Exp}(\lambda)\sim \textrm{Gamma}(1,\lambda)$.
– Zen
Mar 27 '12 at 17:29
• That answers everything very precisely. I was going to nitpick about rate and scaling parameters, but Dilip beat me to it. In the long run, that really doesn't matter; the rate parameter makes a bit more sense anyways. Thanks, all!
– Poik
Mar 27 '12 at 17:40