Suppose a random variable has a frequency distribution that is Poisson with parameter . Suppose the parameter is also a random variable and it has a gamma distribution with parameters and . Then is equivalent to a negative binomial with parameters and .

Note that

- When , the gamma distribution is equivalent to an exponential distribution.
- This also means the negative binomial has parameter 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

per year. The parameter

varies by bot according to an exponential distribution with parameter

.

- What is the expected number of discoveries per year for a bot chosen at random?

Answer: 3
- What is the variance?

Answer: 12

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