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?
    Answer:  3
  2. What is the variance?
    Answer:  12
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