IAI Actuarial Core Principles · Actuarial Statistics

Basic Univariate Distributions and Generating Samples

This chapter covers the standard discrete and continuous distributions you use to model single random quantities, and the methods to simulate values from them. To solve questions, recall the pdf, mean and variance, then apply the inverse transform or acceptance-rejection method and judge accuracy using the standard error.

What this chapter covers

This chapter builds your toolkit of single-variable distributions. You learn the binomial, Poisson and geometric distributions for counts, and the normal, lognormal, exponential and gamma distributions for amounts and waiting times. You also learn the uniform distribution, which is the base for almost all simulation.

The second half is about generating samples. With the inverse transform method you turn a U(0,1) value into a value from a target distribution using its inverse CDF. With acceptance-rejection you handle distributions whose CDF cannot be inverted easily. You finish by asking how many simulations you need for a given accuracy.

These ideas feed the rest of the paper. Statistical inference, regression and Bayesian work all assume you know these distributions well. CS2 uses them for claim counts, claim sizes, waiting times and Monte Carlo methods. The computer-based Paper B also expects you to generate and check samples in R, so the theory needs to be clear enough to code.

Almost every later chapter assumes you can write down a distribution, state its mean and variance, and recognise which one fits a situation. Multiple-choice questions often test these facts directly, and written questions on simulation reward clear method: inverse CDF derived correctly, steps shown, and assumptions stated. The same material supports Paper B, where you must turn the method into working R code. Time spent here pays off across the whole paper, because errors in basics carry into every later answer.

Basic univariate distributions and generating samples: topics in the order to study them

1Discrete Distributions: Binomial, Poisson, GeometricStart with counts. Their formulas are simple, and they set the habit of stating pmf, mean, variance and conditions.
2Continuous Distributions: Normal, Lognormal, Exponential, GammaNext move to continuous models. You need these pdfs, moments and links (for example, exponential as a gamma case) before simulating them.
3Uniform Distribution and Distribution PropertiesThe uniform is the source of all random numbers. Properties such as the CDF, the quantile and moment results are needed for the inverse transform.
4Generating Random Samples: Inverse Transform MethodThis is the core simulation method. It uses the CDF and quantile ideas you have just revised.
5Acceptance-Rejection and Other Simulation MethodsLearn this after inverse transform, because it is the fallback when the CDF cannot be inverted in closed form.
6Simulation Accuracy and Number of SimulationsFinish with accuracy. It uses the sample mean, standard error and the normal distribution, so it comes last.

How to prepare Basic univariate distributions and generating samples

Aim to move from recalling facts to applying methods under time pressure. Short, frequent sessions work well if you study alongside work.

Make one sheet per distribution with its support, pmf or pdf, mean, variance and a typical use. Rewrite it from memory until it is error-free.
Practise recognising the distribution from a description, such as number of trials until the first success or time between events.
Derive the CDF and inverse CDF yourself for the exponential and other simple cases. Do not just memorise the results.
Do simulation questions step by step: state the U(0,1) value, apply the inverse CDF, and write the sample value.
For acceptance-rejection, practise choosing the envelope, finding the constant and writing the acceptance condition and the steps clearly.
Work out the standard error and the number of simulations needed for a stated accuracy, and check the units and the confidence level.
Repeat key simulations in R so you can generate, plot and check samples in Paper B, and then do timed past-paper style questions.

Common mistakes in Basic univariate distributions and generating samples

Mixing up the two definitions of the geometric distribution.
Fix: Read the question for the definition and the support. Write the support (starting at 0 or 1) before you use any formula.
Using the wrong parameters for the normal, lognormal or gamma distributions.
Fix: Write the parameterisation first. For the lognormal, remember that μ and σ² belong to ln X, not X. Check the mean formula against it.
Applying the inverse transform without deriving the inverse CDF correctly.
Fix: Set u = F(x), solve for x, and test the result at u = 0 and u = 1 to confirm it lies in the support.
Choosing an acceptance-rejection constant that is too small, or one that does not bound f/g.
Fix: Find the maximum of f(x)/g(x) by calculus or by inspection, and set c to that value, so that f ≤ c·g everywhere.
Misreading simulation accuracy.
Fix: Use standard error = s ÷ √n. Then build the confidence interval or solve for n with the right normal critical value.
Giving a numeric answer without showing method or assumptions.
Fix: Write the formula in standard notation, the substitution, the result and one line about any assumptions, such as independence of the simulated values.

Last-day revision: Basic univariate distributions and generating samples

Binomial(n, p): mean np, variance np(1 − p).
Poisson(λ): mean = variance = λ.
Geometric: counts trials or failures until the first success, so check which definition the question uses.
Exponential(λ): mean 1/λ, variance 1/λ², CDF F(x) = 1 − e^(−λx).
Gamma(α, λ) with α = 1 is the exponential distribution.
If X is lognormal, then ln X is normal.
Inverse transform: if U ~ U(0,1), then X = F⁻¹(U) has CDF F.
For the exponential, X = −ln(1 − U) ÷ λ, and −ln(U) ÷ λ works too since 1 − U is also U(0,1).
Acceptance-rejection needs a density g and a constant c with f(x) ≤ c·g(x) for all x.
The acceptance probability is 1/c, so a smaller c is more efficient.
Standard error of a simulated mean = s ÷ √n.
To halve the standard error, you need four times as many simulations.

Basic univariate distributions and generating samples practice questions

Basic univariate distributions and generating samples in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Basic univariate distributions and generating samples: frequently asked questions

Which distributions should I know best in this chapter?

Know the binomial, Poisson, geometric, normal, lognormal, exponential, gamma and uniform distributions. For each, be able to state the support, the pmf or pdf, the mean and the variance. You should also recognise which situation each one models.

How do I use the inverse transform method?

Find the CDF F(x), set u = F(x) and solve for x to get F⁻¹(u). Then generate U from U(0,1) and compute X = F⁻¹(U). The resulting X has the distribution with CDF F.

When should I use acceptance-rejection instead?

Use it when the CDF cannot be inverted in closed form, as with some gamma or normal-type densities. You pick a simpler density g that you can simulate from, and a constant c with f(x) ≤ c·g(x). You then accept each candidate with probability f(x) ÷ (c·g(x)).

How many simulations do I need?

It depends on the accuracy you want. The standard error of a simulated mean is s ÷ √n, so you solve for n using the target error and the confidence level. Because of the square root, four times as many simulations only halves the error.

Do I need R for this chapter?

The theory appears in Paper A and the computer-based Paper B tests practical skills. You should be able to generate samples, apply the inverse transform and check results in R. Practise the same methods by hand and in code.