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IAI Actuarial Core Principles · Actuarial Statistics

Generating Functions for IAI Actuarial Statistics

A generating function packs a whole distribution into one function. The MGF is M(t) = E[e^(tX)], and its derivatives at 0 give moments. The CGF is ln M(t) and gives mean and variance. The PGF is E[s^N] for counts. For independent sums, you multiply the functions.

What this chapter covers

This chapter covers generating functions: the moment generating function (MGF), the cumulant generating function (CGF) and the probability generating function (PGF). Each one encodes a distribution in a single function. You use it to find moments, to identify a distribution, and to handle sums of random variables.

The main idea is simple. For independent X and Y, M(X+Y)(t) = M_X(t) × M_Y(t). Adding random variables becomes multiplying functions. This is much easier than convolution. The same idea carries over to the PGF for counting variables, and it leads to compound distributions such as S = X₁ + X₂ + ... + X_N with random N.

This chapter connects to the rest of the paper. You use it in the distributions chapters, in the central limit theorem and in the aggregate claims models of risk modelling. It also supports later work in CS2 on compound Poisson processes and aggregate loss. If you learn it well here, many later derivations become short.

Generating function questions are compact, and they reward method. A typical question asks you to derive an MGF, find a mean and variance from it, or identify a distribution from its MGF. Written questions often give full marks for correct steps, so a clean derivation is worth the effort. The same tools appear in multiple-choice questions, where a quick identification saves time. The chapter also lifts your accuracy in other chapters, because sums and compound models rely on it.

Generating functions: topics in the order to study them

  1. 1Moment Generating Function (MGF) BasicsEverything else builds on the definition M(t) = E[e^(tX)] and on how derivatives at 0 give moments.
  2. 2MGFs of Standard DistributionsOnce you know the definition, you practise it on the normal, exponential, gamma, Poisson, binomial and others, which you will need in the exam.
  3. 3Cumulant Generating FunctionThe CGF is ln M(t), so you need the MGF first; it gives mean and variance with less algebra.
  4. 4Probability Generating Function (PGF)The PGF is the counting-variable version of the same idea, and it is easier once you are comfortable with the MGF.
  5. 5Sums of Random Variables and Compound DistributionsThis topic uses all the earlier tools, so you study it last, when you can combine them.

How to prepare Generating functions

Treat this chapter as a short toolkit that you must be able to use fast. Learn the definitions first, then drill the standard results, then apply them to sums.

  1. Write the definitions from memory: M(t) = E[e^(tX)], K(t) = ln M(t), G(s) = E[s^N]. Check the conditions under which each exists.
  2. Derive the MGF of the exponential, Poisson and normal yourself, step by step, until you can do it without notes. Then memorise the rest from a one-page sheet.
  3. Practise getting moments: E[X] = M′(0) and E[X²] = M″(0). Then get the same answer from the CGF using K′(0) and K″(0), and compare.
  4. For the PGF, practise finding probabilities from coefficients, and the mean as G′(1). Also find the variance as G″(1) + G′(1) − [G′(1)]².
  5. Solve sums of independent variables by multiplying MGFs, then identify the result. Do at least five such questions.
  6. Work through compound distributions, writing S = X₁ + ... + X_N clearly. Use E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)(E[X])², and check each assumption.
  7. Finish with timed past-paper questions. Write the full working, not just the final value.

Common mistakes in Generating functions

  • Adding MGFs instead of multiplying them for a sum of independent variables.

    Fix: Say it as a rule: sum of variables means product of MGFs, and only when the variables are independent.

  • Forgetting to evaluate derivatives at t = 0 (or at s = 1 for the PGF).

    Fix: Write the evaluation point in your working every time, for example M′(0), and give a number as the final answer.

  • Using the wrong variance formula from the PGF.

    Fix: Remember that G″(1) = E[N(N − 1)], so Var(N) = G″(1) + G′(1) − [G′(1)]².

  • Using the wrong parameterisation of the exponential or gamma.

    Fix: State the parameterisation before you write the MGF and check the condition on t, such as t < λ.

  • Applying the compound formulas when X's are not independent of N or of each other.

    Fix: Write the assumptions: the X's are independent and identically distributed, and independent of N.

  • Skipping the working in written questions.

    Fix: Show the definition, the substitution and the derivative steps. Marks are given for method as well as the result.

Last-day revision: Generating functions

  • MGF: M(t) = E[e^(tX)], valid where the expectation is finite for t near 0.
  • E[Xʳ] = the r-th derivative of M(t) at t = 0.
  • Independent X and Y: M(X+Y)(t) = M_X(t) M_Y(t).
  • For Y = aX + b: M_Y(t) = e^(bt) M_X(at).
  • CGF: K(t) = ln M(t); K′(0) = mean and K″(0) = variance.
  • Exponential with rate λ: M(t) = λ ÷ (λ − t), for t < λ.
  • Poisson with mean λ: M(t) = exp(λ(eᵗ − 1)).
  • Normal(μ, σ²): M(t) = exp(μt + σ²t²÷2).
  • PGF: G(s) = E[s^N]; G(1) = 1 and E[N] = G′(1).
  • Var(N) = G″(1) + G′(1) − [G′(1)]².
  • Compound sum S: E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)(E[X])².
  • If the MGF exists near 0, it identifies the distribution uniquely.

Generating functions practice questions

Generating functions in other exams

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

Generating functions: frequently asked questions

Do I need to derive MGFs in the exam?

Be ready to. Simple ones, such as the exponential and Poisson, are often asked as derivations. For the others, know the result and the condition on t.

When should I use the CGF rather than the MGF?

Use the CGF when you want the mean and variance quickly. Differentiating ln M(t) is often shorter than differentiating M(t) twice. For independent sums, CGFs also add.

How is the PGF different from the MGF?

The PGF is defined for non-negative integer-valued variables as G(s) = E[s^N]. The MGF works for any variable where the expectation exists. They are linked by M(t) = G(eᵗ).

How do I identify a distribution from its MGF?

Match the form of the MGF to a known standard one and read off the parameters. This works because an MGF that exists near 0 determines the distribution uniquely.