Frequency models count the number of times an event occurs.

- The number of customers to arrive each hour.
- The number of coins lucky Tom finds on his way home from school.
- How many scientists a Tyrannosaur eats on a certain day.
- Etc.

This is in contrast to a severity model which measures the magnitude of an event.

- How much a customer spends.
- The value of a coin that lucky Tom finds.
- The number of calories each scientist provides.
- Etc.

The following distributions are used to model event frequency. For notation, means .

##
Poisson:

Properties:

- Parameter is .
- Mean is .
- Variance is .
- If are Poisson with parameters , then is Poisson with parameter .

##
Negative Binomial:

Properties:

- Parameters are and .
- Mean is .
- Variance is .
- Variance is always greater than the mean.
- Is equal to a Geometric distribution when .
- If are negative binomial with parameters and , then the sum is negative binomial and has parameters and . Note: ‘s must be the same.

##
Geometric:

Properties:

- Parameter is .
- Mean is .
- Variance is .
- If are geometric with parameter , then the sum is negative binomial with parameters and .

##
Binomial:

where is a positive integer, .

Properties:

- Parameters are and .
- Mean is .
- Variance is .
- Variance is always less than mean.
- If is binomial with parameters and , then the sum is binomial with parameters and .

##
The (a,b,0) recursion:

These distributions can be reparameterized into a recursive formula with parameters and . When reparameterized, they all have the same recursive format.

It is more common to write

The parameters and are different for each distribution.

**Poisson**:

and .**Negative Binomial**:

and .**Geometric**:

and .**Binomial**:

and .

**Pop Quiz!**

- A frequency distribution has a = 0.8 and b = 1.2. What distribution is this?

Answer: Negative Binomial because both parameters are positive. - A frequency distribution has mean 1 and variance 0.5. What distribution is this?

Answer: Binomial because the variance is less than the mean.

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