Recognizing (a,b,0) and (a,b,1) Distributions | Free Exam FAM Lesson

A recursion ratio is $0.3 + 0.6/ k$ . Is this Poisson? Negative binomial? Logarithmic? Recognizing which distribution you have (and how it relates to its zero-truncated siblings) separates a 30-second solution from a 5-minute grind.

Poisson( $λ$ ): $a = 0$ , $b = λ$ . Recursion ratio $λ / k$ , decreasing.

Binomial( $m, q$ ): $a = - q / (1 - q)$ , $b = (m + 1) q / (1 - q)$ . Support ${0, \dots, m}$ , finite.

Negative binomial( $r, β$ ): $a = β / (1 + β)$ , $b = (r - 1) β / (1 + β)$ . Infinite support. Tail decays geometrically.

Geometric( $β$ ): NB with $r = 1$ . $b = 0$ . Constant recursion ratio $a$ .

KEY: Geometric is negative binomial with $r = 1$ , giving $b = 0$ . The recursion ratio is constant at $a$ for all $k$ , if the ratio does not change with $k$ , you have geometric.

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Common mistakes

Confusing zero-truncated with logarithmic. Both have $p_{0} = 0$ . But logarithmic is NOT the ZT version of any (a,b,0) distribution. Its $a + b = 0$ signature is distinct. Trap: computing $r = b / a + 1 = 0$ , which is not a valid NB parameter.
Starting the (a,b,1) recursion from $p_{0}$ . The recursion starts from $p_{1}$ . Computing $p_{1} = (a + b) \times 0 = 0$ makes everything vanish. Trap: all probabilities zero.
Forgetting ZT rescaling. $p_{k}^{T} = p_{k} / (1 - p_{0})$ . When $p_{0}$ is large (Poisson with small $λ$ ), the rescaling is substantial. Trap: reporting un-rescaled probabilities.

Bottom line

(a,b,0) has exactly four members: Poisson, binomial, negative binomial, geometric
(a,b,1) adds all zero-truncated/zero-modified versions, plus logarithmic ( $a + b = 0$ )
Logarithmic has $a + b = 0$ , $a > 0$ , no mass at zero, and no (a,b,0) counterpart
ZT rescaling: $p_{k}^{T} = p_{k} / (1 - p_{0})$ . Same recursion factors, different starting value

Exam shortcut

Build a decision tree. First: mass at zero? No → (a,b,1). Second: $a + b = 0$ ? Yes → logarithmic. No → ZT or ZM version. Third: sign of $a$ identifies the underlying family. Three steps, under 15 seconds. "Logarithmic = a plus b equals zero." Drill as reflex. "ZT = divide by 1 minus p-naught." One formula covers all four ZT versions. "The sign tells the story."

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Learning objectives

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