Welcome to the World of Frequency!
In the "Severity, Frequency, and Aggregate Models" section of Exam FAM, we spend a lot of time counting things—specifically, how many claims occur in a given period. This is what we call Frequency. Instead of memorizing dozens of unrelated formulas, actuaries use two "families" or classes of distributions: the (a, b, 0) class and the (a, b, 1) class.
Think of these classes like a "template." If a distribution fits the template, we can use a standard set of rules to find its probabilities. This chapter is your guide to mastering these templates. Don't worry if the math looks a bit intimidating at first; we’ll break it down step-by-step!
1. The Fundamental Building Block: The (a, b, 0) Class
The (a, b, 0) class is a group of three famous discrete distributions: the Poisson, the Binomial, and the Negative Binomial. What makes them a "family" is that the ratio of successive probabilities follows a very specific linear pattern.
The Magic Formula
For any distribution in the (a, b, 0) class, the probability of having \(k\) claims (\(p_k\)) is related to the probability of having \(k-1\) claims (\(p_{k-1}\)) by this formula:
\(\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \quad k = 1, 2, 3, ...\)
Wait, what does this mean? Imagine you are climbing a ladder. Each step (\(p_k\)) is related to the step below it (\(p_{k-1}\)) by a simple multiplier that changes slightly as you go higher (\(k\)). The "0" in (a, b, 0) tells us that this rule starts working right from the very first step, starting with \(p_0\).
Meet the Family Members
There are only three distributions that fit this rule. Here is how to identify their a and b parameters:
- Poisson (\(\lambda\)):
\(a = 0\)
\(b = \lambda\)
Key trait: Since \(a=0\), the ratio is just \(\lambda/k\). - Binomial (\(m, q\)):
\(a = \frac{-q}{1-q}\)
\(b = \frac{(m+1)q}{1-q}\)
Key trait: \(a\) is always negative here. - Negative Binomial (\(r, \beta\)):
\(a = \frac{\beta}{1+\beta}\)
\(b = \frac{(r-1)\beta}{1+\beta}\)
Key trait: \(a\) is always positive and less than 1.
Quick Review: How do you know which distribution you're looking at?
- If \(a < 0\): It’s Binomial.
- If \(a = 0\): It’s Poisson.
- If \(a > 0\): It’s Negative Binomial.
Did you know? The Geometric distribution is actually just a Negative Binomial where \(r = 1\). This means for a Geometric distribution, \(b\) will always be 0!
Summary Key Takeaway: The (a, b, 0) class uses a simple recursive ratio to find probabilities. Memorizing the relationship between \((a, b)\) and the distribution parameters (\(\lambda, m, q, r, \beta\)) is essential for speed on the exam.
2. Leveling Up: The (a, b, 1) Class
Sometimes, real-world data doesn't perfectly fit the (a, b, 0) distributions. The most common problem? The number of "zero-claim" outcomes is either much higher or much lower than the model predicts. This is where the (a, b, 1) class comes in.
What’s the difference?
The (a, b, 1) class uses the exact same recursion formula: \(\frac{p_k}{p_{k-1}} = a + \frac{b}{k}\). However, it only starts working for \(k = 2, 3, ...\). The probability for zero (\(p_0\)) is "broken" or "modified" and doesn't follow the rule.
There are two types of distributions in this class:
- Zero-Truncated (ZT): These distributions have no zeros at all (\(p_0 = 0\)). Think of a hospital record of patients; you can't be a patient if you had zero visits!
- Zero-Modified (ZM): These distributions have a custom zero probability (\(p_0^M\)) that you choose yourself.
How to calculate (a, b, 1) Probabilities
To find probabilities for the (a, b, 1) class, we first look at the "underlying" (a, b, 0) distribution (let's call its probabilities \(p_k\)) and then adjust them.
For Zero-Truncated (ZT):
\(p_k^T = \frac{p_k}{1 - p_0}\)
Explanation: We take the original probability and "stretch" it to fill the gap left by the missing zero.
For Zero-Modified (ZM):
\(p_k^M = \frac{1 - p_0^M}{1 - p_0} p_k\)
Pro-Tip: Another way to think of this is \(p_k^M = (1 - p_0^M) p_k^T\). You are basically saying, "I'll set aside \(p_0^M\) for the zeros, and then distribute the remaining probability among the non-zero outcomes."
Common Mistake to Avoid: When calculating the ratio for an (a, b, 1) distribution, remember that the first ratio you can calculate is \(\frac{p_2}{p_1}\). You cannot use the formula to find \(p_1\) from \(p_0\) because the rule doesn't apply at \(k=1\)!
Summary Key Takeaway: The (a, b, 1) class is just the (a, b, 0) class with a "special" probability at zero. Use the ZT or ZM formulas to adjust the standard probabilities.
3. Putting it Together: Step-by-Step Problem Solving
When you see a problem asking for parameters or probabilities in these classes, follow these steps:
Step 1: Identify the Class
Does the problem mention \(p_0\), \(p_1\), and \(p_2\)? Check the ratio. If \(\frac{p_1}{p_0}\) and \(\frac{p_2}{p_1}\) follow the same \(a + b/k\) rule, it's (a, b, 0). If only the later ones follow it, it's (a, b, 1).
Step 2: Solve for a and b
If you have two ratios, you have two equations:
1. \(\frac{p_k}{p_{k-1}} = a + \frac{b}{k}\)
2. \(\frac{p_{k+1}}{p_{k}} = a + \frac{b}{k+1}\)
Subtract them or use substitution to find \(a\) and \(b\).
Step 3: Identify the Underlying Distribution
Once you have \(a\), use our "Key Trait" list:
- Positive \(a\)? Negative Binomial.
- Zero \(a\)? Poisson.
- Negative \(a\)? Binomial.
Step 4: Answer the Question
The problem might ask for the mean, the variance, or a specific probability. Now that you know the distribution and its parameters (like \(\lambda\) or \(\beta\)), you can use standard formulas to finish.
Don't worry if this seems tricky at first! The most common challenge is the algebra involved in solving for \(a\) and \(b\). Practice solving a few sets of these equations, and it will become second nature.
4. Quick Review Box
The (a, b, 0) Table Summary:
- Poisson: \(a=0\), \(b=\lambda\), \(p_k/p_{k-1} = \lambda/k\)
- Binomial: \(a < 0\), \(a = -q/(1-q)\)
- Neg. Binomial: \(a > 0\), \(a = \beta/(1+\beta)\)
The (a, b, 1) Formula:
\(p_k^M = \frac{1 - p_0^M}{1 - p_0} p_k\)
Key Vocabulary:
- Zero-Truncated: \(p_0\) is exactly 0.
- Zero-Modified: \(p_0\) is a specific value given in the problem.
You've got this! Understanding the relationship between these parameters is a huge step toward passing Exam FAM. Keep practicing those ratios!