Creating new distributions: multiplication, powers, exponentiation, mixing, and splicing

Exam ASTAM – Advanced Short-Term Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the Lab: Creating New Distributions!

In your actuarial journey, you’ve met the "celebrity" distributions: the Exponential, the Gamma, and the Pareto. But real-world insurance data is messy. Sometimes, a standard distribution just doesn't fit the claims data perfectly.

Think of this chapter like a DIY workshop. Instead of buying a new distribution, we take the ones we already have and modify them—stretching them, powering them up, or even stitching them together like a patchwork quilt. By the end of these notes, you’ll know how to "engineer" a distribution to fit almost any severity model scenario.

Quick Review: Remember that a Severity Model describes the size of a loss. In this chapter, we are looking for ways to create new CDFs ($F(x)$) and PDFs ($f(x)$) to better represent those loss sizes.

1. Multiplication (Scaling the Distribution)

This is the simplest way to create a new distribution. If you have a random variable $X$ (representing a claim) and you multiply it by a constant $c > 0$, you get a new random variable $Y = cX$.

Why do we do this?
Imagine you have a distribution for claims in U.S. Dollars, but you need it in Euros. Or, imagine every claim increases by 5% due to inflation. You are scaling the distribution.

The Math:
If $Y = cX$, then the new Cumulative Distribution Function (CDF) is:
$F_Y(y) = F_X(y/c)$

The new Probability Density Function (PDF) is:
$f_Y(y) = \frac{1}{c} f_X(y/c)$

Key Concept: The Scale Parameter ($\theta$)
In many SOA distributions, the parameter $\theta$ is a scale parameter. If you multiply the whole distribution by $c$, you usually just multiply the old $\theta$ by $c$ to get the new one. For example, if $X \sim \text{Exponential}(\theta)$, then $cX \sim \text{Exponential}(c\theta)$.

Common Mistake to Avoid:
Don't forget to divide by $c$ inside the function and multiply by $1/c$ outside when finding the PDF!

Key Takeaway: Multiplying a distribution by a constant $c$ "stretches" the distribution. The shape stays similar, but the scale changes.

2. Power Transformations

What if we take a distribution and raise it to a power? If $X$ is our original variable, we can create $Y = X^{1/\tau}$ (or $Y = X^\tau$).

The Math:
If $Y = X^{1/\tau}$, then $X = Y^\tau$. The CDF becomes:
$F_Y(y) = F_X(y^\tau)$

The "Transformed" Family:
This is how we get some of the most famous distributions in the ASTAM tables:
1. Transformed Gamma: Created by taking a Gamma distribution and raising it to a power.
2. Burr Distribution: Created by taking a Pareto distribution and raising it to a power.

Memory Aid:
Think of the power $\tau$ (tau) as "Toughness." Raising a claim to a power changes how "heavy" the tail of the distribution is. A higher power can make the probability of large claims much higher or lower.

Key Takeaway: Power transformations change the shape and tail weight of the distribution, making it more or less likely to see extreme values.

3. Exponentiation

In this transformation, we take a random variable $X$ and turn it into $Y = e^X$.

The Famous Example: The Lognormal
If $X$ follows a Normal distribution, then $Y = e^X$ follows a Lognormal distribution. This is one of the most important distributions for severity because it is always positive and has a right-skewed tail (perfect for insurance claims!).

The Math:
If $Y = e^X$, then $X = \ln(Y)$. The CDF is:
$F_Y(y) = F_X(\ln y)$

Did you know?
We use the "Log-" prefix for the new distribution (e.g., Loglogistic, Lognormal). It means that if you take the logarithm of your claim size, it will follow the original distribution.

Key Takeaway: Exponentiation is used to create distributions that are strictly positive and often have very long, "heavy" tails.

4. Mixing Distributions

Mixing is like making a smoothie. You aren't adding the claim amounts together; you are saying that a claim has a certain probability of coming from "Distribution A" and a certain probability of coming from "Distribution B."

Discrete Mixtures:
Suppose you have two types of drivers: "Safe" (Distribution $F_1$) and "Risky" (Distribution $F_2$). If 70% of your drivers are safe and 30% are risky, the total distribution is:
$F(x) = 0.70 F_1(x) + 0.30 F_2(x)$

General Formula:
$F(x) = \sum_{i=1}^{n} a_i F_i(x)$, where the weights $a_i$ must sum to 1 ($\sum a_i = 1$).

Step-by-Step for Mixing:
1. Identify the individual distributions ($F_1, F_2, ...$).
2. Identify the weights (the probability of being in each group).
3. Multiply each CDF or PDF by its weight.
4. Add them up!

Common Mistake:
Students often confuse "Mixing" with "Adding Random Variables."
- Mixing: "The claim is either from Type A OR Type B."
- Adding: "The total claim is the SUM of Claim A AND Claim B."
These are mathematically very different! Mixing uses weights; adding involves convolutions (which are much harder).

Key Takeaway: Mixing allows you to model a population that has distinct subgroups with different behaviors.

5. Splicing Distributions

Splicing is the "Frankenstein" method. You use one distribution for small claims and a different one for large claims. This is very common in insurance because small "fender bender" claims behave differently than "total loss" catastrophes.

Analogy:
Imagine a store that sells items for $1 to $10 using one price list, and items over $10 using a completely different price list. The point where the rules change ($10) is called the threshold or join point.

The Math:
A spliced PDF looks like this:
$f(x) = \begin{cases} a_1 f_1(x) & 0 < x < k \\ a_2 f_2(x) & x > k \end{cases}$

Crucial Rule: The Area must equal 1!
The most important thing to remember is that the total area under the spliced PDF must be 1. You usually have to "scale" the individual pieces using weights ($a_i$) so they fit together properly.

Example of Splicing:
You might use an Exponential distribution for claims between $0 and $5,000 (the "body") and a Pareto distribution for claims above $5,000 (the "tail").

Quick Review: How to handle Splicing
- Check the thresholds ($k$).
- Ensure the sum of probabilities across all intervals equals 1.
- If asked for the CDF, integrate each piece carefully, carrying over the probability from the previous piece.

Key Takeaway: Splicing lets you "stitch" different models together to capture different behaviors at different claim sizes.

Final Summary: The Toolbox

Don't worry if these transformations feel abstract. Just remember what each tool does to the "shape" of your data:

- Multiplication: Changes the scale (Currency/Inflation).
- Powers: Changes the tail weight (Burr, Transformed Gamma).
- Exponentiation: Creates strictly positive distributions (Lognormal).
- Mixing: Combines different groups of people (Safe vs. Risky).
- Splicing: Combines different claim sizes (Small vs. Large).

You've got this! These tools are the foundation for building the sophisticated models you'll use throughout the rest of Exam ASTAM.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

If a random variable $X$ is multiplied by a constant $c > 0$ to create $Y = cX$, what is the resulting CDF $F_Y(y)$?

$F_Y(y) = F_X(y/c)$

How is the PDF of a scaled random variable $Y = cX$ calculated from the original PDF $f_X$?

$f_Y(y) = \frac{1}{c} f_X(y/c)$

What happens to the scale parameter $\theta$ of a distribution when the random variable is multiplied by a constant $c$?

The new scale parameter becomes $c\theta$.

If $Y = X^{1/\tau}$, how do you determine the CDF of the transformed variable $Y$?

$F_Y(y) = F_X(y^\tau)$

What is the primary effect of a power transformation on the characteristics of a distribution?

It changes the shape and the tail weight of the distribution.

If $X$ follows a Normal distribution and we create $Y = e^X$, what is the resulting distribution called?

The Lognormal distribution.

What is the general formula for the CDF of a discrete mixture of distributions?

$F(x) = \sum_{i=1}^{n} a_i F_i(x)$, where $\sum a_i = 1$.

What is the conceptual difference between "Mixing" distributions and "Adding" random variables?

Mixing represents an "OR" scenario (one claim from one group), while adding represents an "AND" scenario (the sum of multiple claims).

How is a spliced distribution defined mathematically using a threshold $k$?

$f(x) = a_1 f_1(x)$ for $x < k$ and $f(x) = a_2 f_2(x)$ for $x > k$.

What is the critical constraint when defining the weights of a spliced PDF?

The total area under the entire spliced PDF must equal 1.

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