# The Poisson Gamma Mixture Pattern

Suppose a random variable $N$ has a frequency distribution that is Poisson with parameter $\lambda$. Suppose the parameter $\lambda$ is also a random variable and it has a gamma distribution with parameters $\alpha$ and $\theta$. Then $N$ is equivalent to a negative binomial with parameters $r = \alpha$ and $\beta = \theta$.

Note that

1. When $\alpha =1$, the gamma distribution is equivalent to an exponential distribution.
2. This also means the negative binomial has parameter $r=1$ which is equivalent to a geometric distribution.
Pop Quiz!
You own a space mining company and have sent several exploration bots to scout possible mineral rich asteroids.  Each bot discovers pockets of valuable resources on different asteroids at a rate of $\lambda$ per year.  The parameter $\lambda$ varies by bot according to an exponential distribution with parameter $\theta = 3$.
1. What is the expected number of discoveries per year for a bot chosen at random?
2. What is the variance?