Calculations for the (a,b,0) and (a,b,1) Classes | Free Exam FAM Lesson

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.

HIGH-FREQUENCY: Exam problems give $a$ , $b$ , and $p_{0}$ , then ask for $p_{k}$ at a specific $k$ .

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

Procedure: identify distribution (sign of $a$ ), extract parameters, compute $p_{0}$ , iterate forward.

At $k = 1$ : $p_{1} = (a + b) \cdot p_{0}$ . This shortcut avoids fractions.

KEY: At $k = 1$ , the recursion factor simplifies to $a + b$ . This is always a clean number, use it as a sanity check on your $a$ and $b$ values.

$p_{1}^{T} = \frac{( a + b ) \cdot p _{0}}{1 - p _{0}}$

Then iterate: $p_{k}^{T} = (a + b / k) \cdot p_{k - 1}^{T}$ for $k \geq 2$ . Same factors, different starting value.

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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 $Var / E = 1/ (1 - a)$ , not $E / Var$ . With $a = 1/3$ , ratio is $3/2$ . Inverting gives $2/3$ , computing underdispersed variance for an NB, contradiction.

Bottom line

Iterate: $p_{k} = (a + b / k) \cdot p_{k - 1}$ ; at $k = 1$ the factor is always $a + b$
ZT starting value: $p_{1}^{T} = p_{1} / (1 - p_{0})$ , then apply the same recursion factors from $k = 2$ onward
ZM starting value: $p_{1}^{M} = (1 - p_{0}^{M}) / (1 - p_{0}) \cdot p_{1}$
Variance-to-mean ratio equals $1/ (1 - a)$ : Poisson = 1, binomial < 1, NB > 1

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. "Factor, then multiply." Compute the fraction first, then apply. Never do both in your head. "One over one minus a", the variance-to-mean ratio in one formula.

The full lesson (about 1,744 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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