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.

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:

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

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."

Poisson: $\lambda =b$.

Binomial: $q=-a/(1-a)$, $m=-b/a-1$.

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 1,615 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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