Parameters of the (a,b,0) and (a,b,1) Classes

An auto insurer observes 10,000 policies. Some produce zero claims, a few produce five or more. The (a,b,0) class unifies Poisson, binomial, and negative binomial under one recursive framework. Know the two parameters, and you identify the distribution in seconds.

To identify the role of the parameters, compare the recursion ratio at several claim counts. Its sign pattern identifies the family, while the parameter values determine its mean, variance, and support.

HIGH-FREQUENCY: The (a,b,0) recursion and parameter table are tested constantly. Memorize the table cold.

A counting distribution belongs to the (a,b,0) class if:

$$\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \quad k = 1, 2, 3, \ldots$$

Only three families satisfy this. No other distribution fits.

The sign of $a$ identifies the family immediately:

• $a = 0$: Poisson
• $a < 0$: Binomial
• $a > 0$: Negative binomial (including geometric)

KEY: The sign of $a$ is all you need to identify the distribution family. Zero = Poisson, negative = binomial, positive = negative binomial. Memorize this as "0-Negative-Positive."

Common mistakes

• Confusing the sign of $a$ for binomial. Binomial has $a < 0$. Forgetting the negative sign leads to misidentification. Trap: $q > 1$.
• Mixing up recovery formulas. NB: $r = b/a + 1$. Binomial: $m = -b/a - 1$. These differ by sign and direction. Swapping gives a parameter off by 2.
• Forgetting geometric is NB with $r = 1$. When $b = 0$ and $a > 0$, the distribution is geometric, not "Poisson with $\lambda = 0$."

Bottom line

• Recursion: $p_k/p_{k-1} = a + b/k$. Only Poisson, binomial, and negative binomial (with geometric as a special case) satisfy it.
• Sign of $a$ identifies the family: 0 = Poisson, negative = binomial, positive = negative binomial (or geometric).
• Geometric is NB with $r = 1$, so $b = 0$ and the recursion ratio is constant (memoryless).
• Recovery: Poisson $\lambda = b$; binomial $q = -a/(1-a)$, $m = -b/a - 1$; NB $\beta = a/(1-a)$, $r = b/a + 1$.

Exam shortcut

Read off $a$ and $b$, check the sign of $a$, this identifies the family in under 5 seconds. If $p_0$ is given explicitly, verify it against the formula before proceeding. Mismatch = (a,b,1). "0-Negative-Positive" (Poisson, Binomial, Negative Binomial. "Logarithmic = a plus b equals zero." "BRaG") Beta from a, R from a and b, a identifies family, Geometric when b = 0.

The full lesson (about 1,673 words, 11 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2e

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