Compound Poisson, Binomial and Negative Binomial Distributions

Understand Compound Poisson, Binomial and Negative Binomial

Key rules to remember

Aggregate claims

S = X₁ + X₂ + … + X_N, S = 0 if N = 0

X's are i.i.d. and independent of N.

General mgf of S

M_S(t) = P_N(M_X(t)) = M_N(ln M_X(t))

Valid where M_X(t) exists. P_N(z) = E[z^N].

Mean and variance of S

E[S] = E[N]E[X]; Var(S) = E[N]Var(X) + Var(N)(E[X])²

Holds for any compound distribution under the standard assumptions.

Compound Poisson mgf

M_S(t) = exp{λ(M_X(t) − 1)}

Because P_N(z) = exp{λ(z − 1)} for Poisson(λ).

Compound Poisson moments

E[S] = λE[X]; Var(S) = λE[X²]

Mean and variance of N are both λ. Third central moment is λE[X³].

Compound binomial mgf

M_S(t) = (1 − p + p M_X(t))ⁿ

N ~ Binomial(n, p). Mean np E[X]; variance np Var(X) + np(1 − p)(E[X])².

Compound negative binomial mgf

M_S(t) = (p / (1 − q M_X(t)))^k, with q = 1 − p

N is the number of failures before the kth success, P(N = j) = C(j + k − 1, j) p^k q^j. E[N] = kq/p, Var(N) = kq/p².

Sum of independent compound Poisson

S = S₁ + … + S_m, S_i ~ CP(λ_i, F_i) ⇒ S ~ CP(λ, F), λ = Σλ_i, F(x) = Σ(λ_i/λ)F_i(x)

Requires independence of the S_i. The claim size is a mixture with weights λ_i/λ.

How to solve Compound Poisson, Binomial and Negative Binomial questions

1. 1Write the model and state the assumptions: X's i.i.d., independent of N, S = 0 when N = 0.
2. 2Identify the count distribution and its parameters. Note whether N is Poisson, binomial or negative binomial.
3. 3Write the pgf of N, P_N(z), or the mgf M_N(t).
4. 4Write the claim size mgf M_X(t) if needed, or its moments E[X], E[X²], E[X³].
5. 5Substitute: M_S(t) = P_N(M_X(t)). Simplify.
6. 6For moments, either differentiate M_S(t) at t = 0 or use the standard mean and variance formulas.
7. 7For a sum of independent variables, multiply the mgfs, or add the λ's and form the mixture claim size distribution.
8. 8State the result clearly and check that units and parameter values make sense.

Quickest way: Plug the count pgf into the claim size mgf

Common mistakes in Compound Poisson, Binomial and Negative Binomial

• Writing the compound Poisson variance as λVar(X).

  Students copy the general formula but forget that Var(N) = λ, not 0.

  Fix: Use Var(S) = λE[X²]. Check: λVar(X) + λ(E[X])² = λE[X²].

• Substituting t into the count mgf instead of ln M_X(t).

  Students mix up the mgf and pgf forms of the count distribution.

  Fix: If you use the pgf, substitute M_X(t) for z. If you use the mgf, substitute ln M_X(t) for t.

• Adding the claim size distributions without weights when summing compound Poisson variables.

  Students think the claim size is just F₁ + F₂.

  Fix: The combined claim size is a mixture: F(x) = Σ(λ_i/λ)F_i(x). The weights sum to 1.

• Using the negative binomial formulas with the wrong parameterisation.

  Several versions exist: failures before the kth success, or total trials.

  Fix: Check the definition given in the question or the Tables. Write down E[N], Var(N) and the pgf before you continue.

• Applying the sum result to dependent or non-compound-Poisson components.

  The result is memorised without its conditions.

  Fix: State that the S_i are independent and each is compound Poisson. The result does not hold in general for compound binomial or negative binomial.

• Forgetting that M_X(t) must exist at the t used.

  Students differentiate without checking the domain.

  Fix: Note any restriction on t. For moments you only need the mgf to exist in an interval around 0.

Worked examples