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

Welcome to the World of Claim Frequencies!

In the "Severity, Frequency, and Aggregate Models" section of Exam FAM, we spend a lot of time talking about Frequency—which is just a fancy actuarial way of asking, "How many times did something happen?"

Instead of memorizing dozens of different formulas for every possible distribution, actuaries use two "classes" of distributions called the (a,b,0) class and the (a,b,1) class. These classes allow us to use one simple recursive formula to find probabilities for the most common distributions we see in the real world. Don't worry if this seems a bit mathematical at first; once you see the pattern, it’s like following a recipe!

The Core Secret: The Recursive Formula

The magic of these classes lies in a single relationship between the probability of having \(k\) claims and the probability of having \(k-1\) claims. For any distribution in these classes, the probabilities follow this rule:

\[ \frac{p_k}{p_{k-1}} = a + \frac{b}{k} \]

Where:
\(p_k\) is the probability that the number of claims \(N\) equals \(k\).
\(a\) and \(b\) are constants specific to the distribution.
\(k\) is the number of claims (1, 2, 3...).

Analogy Time: Imagine you are climbing a ladder. The (a,b,0) rule tells you exactly how much effort it takes to get to the next rung based on which rung you are currently standing on. You don't need to jump from the ground to the 10th rung; you just need to know how to get from the 9th to the 10th!

Quick Review Box:
If you know \(p_0\) and you know the values for \(a\) and \(b\), you can find \(p_1\). Once you have \(p_1\), you can find \(p_2\), and so on! It’s a chain reaction.

Section 1: The (a,b,0) Class

The (a,b,0) class is the "base" family. It starts at \(k=0\) and follows the recursive formula for all values of \(k = 1, 2, 3, ...\). There are only three members in this exclusive club:

1. The Poisson Distribution

The Poisson is the bread and butter of frequency. It assumes events happen independently at a constant rate.
For Poisson(\(\lambda\)):
\(a = 0\)
\(b = \lambda\)
\(p_0 = e^{-\lambda}\)

Memory Trick: Poisson starts with "P," and in the (a,b,0) formula, \(a\) is 0. "Poisson has Zero 'a'."

2. The Binomial Distribution

Used when there is a fixed number of trials (\(n\)) and a probability of success (\(q\)).
For Binomial(\(n, q\)):
\(a = \frac{-q}{1-q}\)
\(b = \frac{(n+1)q}{1-q}\)
\(p_0 = (1-q)^n\)

Note: In the Binomial distribution, \(a\) is always negative. This is because the number of possible outcomes is finite (you can't have more than \(n\) claims).

3. The Negative Binomial Distribution

Often used when the claim frequency is more "spread out" (volatile) than a Poisson.
For Negative Binomial(\(r, \beta\)):
\(a = \frac{\beta}{1+\beta}\)
\(b = \frac{(r-1)\beta}{1+\beta}\)
\(p_0 = (1+\beta)^{-r}\)

Summary Table for (a,b,0) Class

Poisson: \(a=0, b=\lambda\)
Binomial: \(a < 0\)
Neg. Binomial: \(a > 0\)

Section 2: The (a,b,1) Class

Sometimes, the "zero" case doesn't fit the pattern. For example, maybe a policy has a deductible so small that zero-claim events are never reported, or maybe there’s a bonus for having zero claims. This is where the (a,b,1) class comes in.

In the (a,b,1) class, the recursive formula still works, but only starting at \(k=2\). The relationship between \(p_0\) and \(p_1\) is "broken" or modified.

There are two types of distributions in this class:

1. Zero-Truncated (ZT) Distributions

These distributions cannot be zero. The probability \(p_0^T = 0\).
If you see a superscript \(T\) (like \(p_k^T\)), it means Truncated.

2. Zero-Modified (ZM) Distributions

These distributions have a custom-assigned probability for zero, which we call \(p_0^M\). The rest of the probabilities (\(p_1, p_2, ...\)) are adjusted to make sure everything still adds up to 100%.

Did you know?
A Zero-Modified distribution is actually just a "weighted average" between a giant spike at zero and a Zero-Truncated distribution!

Calculating Probabilities in the (a,b,1) Class

If you need to find probabilities for a Zero-Modified (\(M\)) distribution, follow these steps:

Step 1: Find the "regular" (a,b,0) probability for zero, which we call \(p_0\).
Step 2: Identify the new modified probability for zero, \(p_0^M\) (usually given in the problem).
Step 3: Calculate the first modified probability \(p_1^M\) using this formula:
\[ p_1^M = \frac{p_0^M - 1}{p_0 - 1} \cdot p_1 \]
Step 4: Use the standard recursive formula (\(p_k = p_{k-1}(a + b/k)\)) to find all subsequent probabilities (\(p_2, p_3, ...\)) based on \(p_1^M\).

Common Mistake: Many students try to use the (a,b,0) recursion to get from \(p_0^M\) to \(p_1^M\). Don't do it! The recursion only works for \(k \ge 2\) in the (a,b,1) class. You must use the "Step 3" formula above to jump from \(p_0\) to \(p_1\).

Key Differences Recap

(a,b,0) Class: Recursion starts at \(k=1\). Includes Poisson, Binomial, and Negative Binomial.
(a,b,1) Class: Recursion starts at \(k=2\). Includes Zero-Truncated and Zero-Modified versions of the above.
The "a" Sign:
- If \(a=0\): It's a Poisson variation.
- If \(a<0\): It's a Binomial variation.
- If \(a>0\): It's a Negative Binomial variation.

Final Encouragement

The (a,b,0) and (a,b,1) classes are simply tools to make your life easier. Instead of learning 10 different probability mass functions, you just need to know how to use one recursive ratio. Practice identifying whether a problem is asking for a "Truncated" (no zeros) or "Modified" (special zero) distribution, and you'll be well on your way to mastering Frequency models!

Key Takeaway: Always check if the distribution is "Zero-Modified" before you start your calculations. If it is, handle \(p_1\) with the special formula first, then let the recursion do the rest of the work!