Frequency Models

Frequency models count the number of times an event occurs.

  1. The number of customers to arrive each hour.
  2. The number of coins lucky Tom finds on his way home from school.
  3. How many scientists a Tyrannosaur eats on a certain day.
  4. Etc.
This is in contrast to a severity model which measures the magnitude of an event.
  1. How much a customer spends.
  2. The value of a coin that lucky Tom finds.
  3. The number of calories each scientist provides.
  4. Etc.
The following distributions are used to model event frequency.  For notation, p_n means Pr(N=n).

Poisson:

\begin{array}{lr}\displaystyle p_n = e^{-\lambda} \frac{\lambda^n}{n!} & \lambda > 0 \end{array}
Properties:
  1. Parameter is \lambda.
  2. Mean is \lambda.
  3. Variance is \lambda.
  4. If N_1, N_2, ..., N_i are Poisson with parameters \lambda_1, \lambda_2, ..., \lambda_i, then N = N_1 + N_2 + ... + N_i is Poisson with parameter \lambda = \lambda_1 + \lambda_2 + ... + \lambda_i.

Negative Binomial:

\begin{array}{lr} \displaystyle p_n = {{n+r-1}\choose{n}}\left(\frac{1}{1+\beta}\right)^r\left(\frac{\beta}{1+\beta}\right)^n & \beta>0, r>0 \end{array}
Properties:
  1. Parameters are r and \beta.
  2. Mean is r\beta.
  3. Variance is r\beta\left(1+\beta\right).
  4. Variance is always greater than the mean.
  5. Is equal to a Geometric distribution when r=1.
  6. If N_1, N_2, ..., N_i are negative binomial with parameters \beta_1 = \beta_2 = ... = \beta_i and r_1, r_2, ..., r_i, then the sum N = N_1 + N_2 + ... + N_i is negative binomial and has parameters \beta = \beta_1 and r = r_1+r_2+...+r_i.  Note: \beta‘s must be the same.

Geometric:

\begin{array}{lr} \displaystyle p_n = \frac{\beta^n}{\left(1+\beta\right)^{n+1}} & \beta>0 \end{array}
Properties:
  1. Parameter is \beta.
  2. Mean is \beta.
  3. Variance is \beta\left(1+\beta\right).
  4. If N_1, N_2, ..., N_i are geometric with parameter \beta, then the sum N = N_1+N_2+...+N_i is negative binomial with parameters \beta and r = i.

Binomial:

\displaystyle p_n = {{m} \choose {n}}q^n\left(1-q\right)^{m-n}
where m is a positive integer, 0<q<1.
Properties:
  1. Parameters are m and q.
  2. Mean is mq.
  3. Variance is mq\left(1-q\right).
  4. Variance is always less than mean.
  5. If N_1, N_2, ..., N_i is binomial with parameters q and m_1, m_2, ..., m_i, then the sum N=N_1+N_2+...+N_i is binomial with parameters q and m = m_1+m_2+...+m_i.

The (a,b,0) recursion:

These distributions can be reparameterized into a recursive formula with parameters a and b.  When reparameterized, they all have the same recursive format.
\displaystyle p_k = \left(a+ \frac{b}{k}\right)p_{k-1}
It is more common to write
\displaystyle \frac{p_k}{p_k-1} = a+\frac{b}{k}
The parameters a and b are different for each distribution.
  1. Poisson:
    a = 0 and b =\lambda.
  2. Negative Binomial:
    \displaystyle a = \frac{\beta}{1+\beta} and \displaystyle b = \left(r-1\right)\frac{\beta}{1+\beta}.
  3. Geometric:
    \displaystyle a = \frac{\beta}{1+\beta} and \displaystyle b = 0.
  4. Binomial:
    \displaystyle a = -\frac{q}{1-q} and \displaystyle b = \left(m+1\right)\frac{q}{1-q}.
Pop Quiz!
  1. A frequency distribution has a = 0.8 and b = 1.2.  What distribution is this?
    Answer: Negative Binomial because both parameters are positive. 
  2. 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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Filed under Frequency Models, Probability

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