Introduction to Continuous Distributions

Welcome to one of the most vital chapters in your CS1 journey! In the previous chapter, we looked at discrete distributions—where we "counted" things (like the number of claims). Now, we move into the continuous world, where we "measure" things. Think about the exact time until the next car accident or the specific amount of an insurance claim. Because these values can be anything (like 124.50 or 124.5023...), we use continuous distributions to model them.

Don't worry if this seems tricky at first! While the math involves some calculus, the core idea is simple: we are looking at the "shape" of data across an interval. By the end of this note, you’ll be familiar with the "actuarial toolkit" of distributions used to predict the future.

Note: For how to calculate moments or use generating functions, see the "Moment and cumulant generating functions" chapter. For the "inverse transform method" to generate these variables, see the "The Poisson process and simulation" chapter.

1. The Fundamentals: PDF and CDF

In the continuous world, the probability of a random variable \(X\) being exactly one specific value is always zero: \(P(X = x) = 0\). Instead, we talk about the probability of \(X\) falling within a range.

  • Probability Density Function (PDF), \(f(x)\): This represents the "height" of the curve. The total area under this curve must equal 1.
  • Cumulative Distribution Function (CDF), \(F(x)\): This is the probability that \(X\) is less than or equal to a value \(x\). Mathematically: \(F(x) = P(X \le x)\).

Quick Tip: To find the probability between two points \(a\) and \(b\), you just calculate \(F(b) - F(a)\).

2. The Uniform Distribution

The Uniform distribution is the simplest continuous distribution. It assumes that every interval of the same length within the range \([a, b]\) is equally likely. Imagine a spinner that is perfectly balanced; it is just as likely to land on 0.1 as it is on 0.9.

Key Properties for \(X \sim U(a, b)\):

  • PDF: \(f(x) = \frac{1}{b-a}\) for \(a \le x \le b\).
  • Mean: \(E(X) = \frac{a+b}{2}\) (Right in the middle!).
  • Variance: \(Var(X) = \frac{(b-a)^2}{12}\).

3. The Exponential Distribution

The Exponential distribution is the "waiting time" distribution. Actuaries use it to model the time between independent events, such as the time between two phone calls to a customer service center.

Key Properties for \(X \sim Exp(\lambda)\):

  • PDF: \(f(x) = \lambda e^{-\lambda x}\) for \(x > 0\).
  • CDF: \(F(x) = 1 - e^{-\lambda x}\).
  • Mean: \(E(X) = \frac{1}{\lambda}\).
  • Variance: \(Var(X) = \frac{1}{\lambda^2}\).

Did you know? The Exponential distribution is "memoryless." This means if you have already waited 10 minutes for a claim to arrive, the probability of waiting another 5 minutes is exactly the same as if you had just started waiting! (Note: This is unique among continuous distributions).

4. The Normal Distribution

The Normal distribution, or the "Bell Curve," is the most famous distribution in statistics. It is symmetrical and defined by its mean (\(\mu\)) and variance (\(\sigma^2\)).

Key Properties for \(X \sim N(\mu, \sigma^2)\):

  • Mean: \(E(X) = \mu\).
  • Variance: \(Var(X) = \sigma^2\).
  • Standardization: We often convert \(X\) to the Standard Normal variable \(Z \sim N(0, 1)\) using the formula: \(Z = \frac{X - \mu}{\sigma}\).

Common Mistake: When using the formula \(Z = \frac{X - \mu}{\sigma}\), remember that \(\sigma\) is the standard deviation (the square root of the variance). Students often forget to take the square root!

5. The Lognormal Distribution

If the natural logarithm of a variable follows a Normal distribution, then the variable itself is Lognormal. It is very popular in insurance for modelling claim sizes because it is "skewed"—meaning it has a long tail to the right for those rare, very expensive claims.

Definition: If \(Y \sim N(\mu, \sigma^2)\), then \(X = e^Y\) is Lognormal.

6. The Gamma Distribution

The Gamma distribution is more flexible than the Exponential. While Exponential models the time until the first event, Gamma can model the time until the \(n\)-th event occurs.

Parameters: It usually has a shape parameter (\(\alpha\)) and a rate parameter (\(\lambda\)).

  • Mean: \(E(X) = \frac{\alpha}{\lambda}\).
  • Variance: \(Var(X) = \frac{\alpha}{\lambda^2}\).

Note: If \(\alpha = 1\), the Gamma distribution is actually just the Exponential distribution!

7. The Beta Distribution

The Beta distribution is unique because it is defined on a fixed interval, usually \([0, 1]\). This makes it perfect for modelling proportions or percentages, like the percentage of a loss that is covered by a specific reinsurance contract.

8. Distributions for Statistical Inference

The following three distributions are primarily used when we are doing "tests" on data. You will see these much more in the Statistical Inference section of the syllabus.

  • Chi-square (\(\chi^2\)): A special case of the Gamma distribution. It is used to test how well a model fits the data (Goodness of Fit).
  • t-distribution: Looks like the Normal distribution but has "fatter tails." We use it when we have small sample sizes and don't know the true population variance.
  • F-distribution: Used primarily to compare two different variances to see if they are significantly different.

9. Evaluating Quantiles and Probabilities

In the CS1 exam, you are expected to find probabilities (like \(P(X < 10)\)) or quantiles (the value \(x\) such that \(P(X < x) = 0.95\)).

Paper A (Manual/Tables)

For Paper A, you will often use the Formulae and Tables book. For the Normal distribution, you look up the "Phi" function \(\Phi(z)\) to find probabilities. For others, you might need to use the CDF formulas directly.

Paper B (Using R)

In the R-based exam (CS1B), you use standard commands. Every distribution has a prefix:

  • p for probability (CDF): e.g., pnorm(x, mean, sd)
  • q for quantile: e.g., qexp(0.95, rate)
  • r for generating random samples: e.g., rgamma(100, shape, rate)

Key Takeaways

  • Continuous variables are measured, not counted.
  • The Normal distribution is our symmetrical benchmark.
  • The Lognormal and Gamma distributions are essential for modelling "skewed" insurance claim data.
  • The Exponential distribution is the go-to for waiting times and has the unique memoryless property.
  • Quantiles and Probabilities can be found using the Formulae and Tables book in Paper A or R functions in Paper B.

Great job! You’ve just covered the essential "cast of characters" in continuous probability. Keep practicing the links between these distributions, as they form the foundation for almost everything else in Actuarial Statistics.