Calculations for the (a,b,0) and (a,b,1) Classes

A reinsurer needs $\Pr(N = 4)$ from a negative binomial. The PMF involves gamma functions and powers. The recursive approach: four multiplications. On a timed exam, the recursion is the only practical path.

Perform calculations sequentially from the supplied starting probability. Each recursion step uses the previous count probability, so skipping a count loses the necessary input.

HIGH-FREQUENCY: Exam problems give $a$, $b$, and $p_0$, then ask for $p_k$ at a specific $k$.

$$p_k = \left(a + \frac{b}{k}\right) p_{k-1}$$

For an (a,b,0) model, identify the distribution, compute its zero probability, and iterate from count 1. For an (a,b,1) model, obtain the positive-count starting value separately and iterate from count 2.

For (a,b,0) only, at $k=1$: $p_1=(a+b)p_0$.

KEY: The first-step factor a plus b applies to (a,b,0) probabilities. It does not connect the zero and one probabilities of a general (a,b,1) model.

$$p_1^T = \frac{(a+b) \cdot p_0}{1 - p_0}$$

Common mistakes

• Forgetting to divide by $1 - p_0$ for ZT. The recursion factors are the same, but the starting value $p_1^T$ must be rescaled. Missing this makes all subsequent probabilities too small. Trap: $p_1^T = 0.2707$ instead of 0.3130.
• Using $p_0$ as starting value in (a,b,1). The recursion starts from $p_1$. If $p_0 = 0$, computing $p_1 = (a+b) \times 0 = 0$ kills the entire sequence.
• Inverting the variance-to-mean ratio. The ratio is $\text{Var}/E = 1/(1-a)$, not $E/\text{Var}$. With $a = 1/3$, ratio is $3/2$. Inverting gives $2/3$, computing underdispersed variance for an NB, contradiction.

Bottom line

• (a,b,0): iterate from $p_0$; the factor at count 1 is $a+b$. For (a,b,1), start from $p_1$ and iterate from count 2.
• ZT starting value: $p_1^T = p_1/(1-p_0)$, then apply the same recursion factors from $k = 2$ onward
• ZM starting value: $p_1^M = (1-p_0^M)/(1-p_0) \cdot p_1$
• For (a,b,0), the variance-to-mean ratio is $1/(1-a)$. Zero truncation or modification changes the moment formulas.

Exam shortcut

Set up a table with columns: $k$, factor $a + b/k$, $p_k$. Fill the factor column first, then multiply down. On problems asking for $\Pr(N \leq 3)$, compute $p_0$ through $p_3$ and sum. Compute each recursion factor before applying it. Check whether the model is (a,b,0) or (a,b,1) before using a starting value or moment formula.

The full lesson (about 1,741 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2g

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