Recognizing (a,b,0) and (a,b,1) Distributions

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.

Recognize each counting family from its recursion coefficients. Their relationships become clearer when you check whether zero truncation or zero modification changes only the probability at zero or the positive probabilities as well.

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

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

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

Geometric($\beta$): 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.

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 $\lambda$), 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) includes zero-truncated/zero-modified versions, logarithmic, and the extended truncated negative binomial with shape between -1 and 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

For an (a,b,1) model, check the zero probability and the recursion from count 2 onward. With positive a, a plus b equal to zero indicates logarithmic positive-count probabilities; a negative sum permits extended truncated negative binomial. Use the supplied zero mass to distinguish truncation from modification. "Logarithmic = a plus b equals zero." Drill as reflex. "ZT = divide by 1 minus p-naught." One formula covers all four ZT versions.

The full lesson (about 1,569 words, 10 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2f

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