Welcome to Selecting Frequency Distributions!

Welcome, future actuaries! So far in your studies, you've likely looked at individual distributions like the Poisson or Negative Binomial. But in the real world (and on Exam FAM), the data doesn't come with a label telling you which distribution to use. This chapter is all about becoming a "distribution detective." We will learn how to look at data characteristics and mathematical patterns to pick the perfect model for the number of insurance claims (frequency). Don't worry if this seems like a lot of math at first—once you see the patterns, it becomes much like a matching game!

1. The "Big Three" Comparison: Mean vs. Variance

The easiest way to start narrowing down your choices is to look at the relationship between the mean \( E[N] \) and the variance \( Var(N) \). In the actuarial world, we call the relationship between these two "dispersion."

Think of it like this: If you expect 10 claims a month, how much does that number swing? Is it always exactly 9, 10, or 11? Or does it swing wildly from 0 to 50?

  • Poisson Distribution (The Balanced Model): Here, the mean equals the variance. \( E[N] = Var(N) = \lambda \). We call this equidispersion.
  • Binomial Distribution (The Consistent Model): Here, the variance is less than the mean. \( Var(N) < E[N] \). This is called underdispersion. This happens when the number of claims is capped (like 10 people in a room, you can't have more than 10 claims).
  • Negative Binomial Distribution (The Volatile Model): Here, the variance is greater than the mean. \( Var(N) > E[N] \). This is called overdispersion. This is very common in insurance because claims often "clump" together.

Quick Review Box:
If Sample Mean \(\approx\) Sample Variance \(\rightarrow\) Poisson
If Sample Mean \(>\) Sample Variance \(\rightarrow\) Binomial
If Sample Mean \(<\) Sample Variance \(\rightarrow\) Negative Binomial

2. The (a, b, 0) Class: The Mathematical Secret Weapon

Actuaries use a special formula to identify distributions within the (a, b, 0) class. This class includes our "Big Three" from above. The rule says that the ratio of successive probabilities follows a linear pattern:

\( \frac{p_k}{p_{k-1}} = a + \frac{b}{k} \) for \( k = 1, 2, 3, \dots \)

By looking at the value of a, you can identify the distribution immediately:

1. Poisson: \( a = 0 \). (The ratio depends only on \( b/k \))
2. Binomial: \( a < 0 \).
3. Negative Binomial: \( a > 0 \).

Analogy Time! Imagine you are walking up a staircase. The height of the next step compared to the current one follows a rule. If the rule is constant (Poisson), the stairs are predictable. If the steps get shorter as you go (Binomial), you'll eventually hit a ceiling. If the steps get taller (Negative Binomial), you're headed for the moon!

Key Takeaway:

If a problem gives you a list of probabilities (like \( p_0, p_1, p_2 \)), calculate the ratio \( \frac{p_k}{p_{k-1}} \). If it decreases as \( k \) increases, but stays positive, check for a. If \( a > 0 \), it's Negative Binomial!

3. Dealing with Zero-Inflation: The (a, b, 1) Class

Sometimes, real-world data has "too many" zeros or "no" zeros at all. For example, if you are looking at "number of hospital stays," you might exclude people with 0 stays (Zero-Truncated) or you might find that way more people have 0 stays than a standard Poisson would suggest (Zero-Modified).

The (a, b, 1) class starts its pattern at \( k = 2 \), meaning \( p_1 \) is calculated differently than \( p_0 \).

  • Zero-Truncated (ZT): The distribution starts at \( k=1 \). There is no probability for \( k=0 \). It's like a club with a "must have 1 claim to enter" sign.
  • Zero-Modified (ZM): We manually set the probability of zero \( p_0^M \) to a specific value, and then "smush" the rest of the distribution to fit the remaining probability.

Don't worry if this seems tricky! The formula for the modified probability \( p_k^M \) is just a scaled version of the original \( p_k \):
\( p_k^M = \frac{1 - p_0^M}{1 - p_0} p_k \)
Where \( p_0^M \) is your new modified zero probability, and \( p_0 \) is the zero probability from the original "Big Three" distribution.

4. Step-by-Step: How to Select a Distribution

When facing an exam question asking you to pick a distribution based on a frequency table, follow these steps:

Step 1: Calculate the sample mean (\( \bar{x} \)) and sample variance (\( s^2 \)).
Step 2: Compare them. If \( s^2 > \bar{x} \), lean toward Negative Binomial. If \( s^2 \approx \bar{x} \), lean toward Poisson.
Step 3: Check the ratios \( k \cdot \frac{p_k}{p_{k-1}} \). If you plot these against \( k \), the slope of the line helps identify a and b.
Step 4: Look at \( p_0 \). If the actual \( p_0 \) in the data is much higher than what your chosen model predicts, you likely need a Zero-Modified version.

Common Mistake to Avoid: Many students forget that the Negative Binomial variance formula is \( r \beta (1+\beta) \). Because of that \( (1+\beta) \) term, the variance will always be larger than the mean \( r \beta \). If your data shows a smaller variance, don't pick Negative Binomial!

5. Graphical Methods (The "Intuition" Check)

Did you know? Actuaries often use a "Ratio Plot" to see which distribution fits. If you plot \( k \frac{p_k}{p_{k-1}} \) against \( k \), and the points form a straight line, it confirms you are looking at an (a, b, 0) or (a, b, 1) distribution.

  • Horizontal Line: Poisson (\( a=0 \)).
  • Upward Sloping Line: Negative Binomial (\( a>0 \)).
  • Downward Sloping Line: Binomial (\( a<0 \)).
Key Takeaway Summary:

Choosing a frequency distribution is about matching patterns. Use the mean-variance relationship for a quick guess, use the \( (a, b, 0) \) recurrence relation for mathematical proof, and use zero-modification if the "0" category looks unusual compared to the rest of the data. Master these three tools, and you'll be the Sherlock Holmes of frequency models!