Sums of Random Variables and Compound Distributions Using MGFs

Understand Sums of Random Variables and Compound Distributions

Key rules to remember

MGF definition

M_X(t) = E[e^(tX)]

Valid for t in an interval around 0 where the expectation is finite.

Sum of independent variables

M_(X₁+…+Xₙ)(t) = M_X₁(t) × … × M_Xₙ(t)

Needs independence. If also identically distributed, the result is [M_X(t)]ⁿ.

Linear transformation

M_(aX+b)(t) = e^(bt) × M_X(at)

Use it for means and scaled variables.

Moments from the MGF

E[Xᵏ] = M_X⁽ᵏ⁾(0)

The kth derivative at t = 0. Variance = M''(0) − [M'(0)]².

PGF of a count

G_N(s) = E[s^N]

G_N(s) = M_N(ln s).

MGF of a compound sum

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

S = X₁ + … + X_N, Xs iid and independent of N.

Compound Poisson MGF

M_S(t) = exp{λ[M_X(t) − 1]}

N ~ Poisson(λ), since G_N(s) = exp{λ(s − 1)}.

Compound mean and variance

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

For compound Poisson, Var(S) = λE[X²].

How to solve Sums of Random Variables and Compound Distributions questions

1. 1Decide whether the sum has a fixed number of terms or a random number. Fixed means multiply MGFs. Random means a compound distribution.
2. 2Check the assumptions: independence between the Xs, and for a compound sum, independence of N from the Xs. State them in your answer.
3. 3Write down the MGF of X (the claim size or the single term) from the standard results.
4. 4For a fixed sum, multiply the MGFs. For a compound sum, write the PGF of N and substitute M_X(t) for s.
5. 5Simplify. Compare with a standard MGF to name the distribution and its parameters.
6. 6If asked for moments, differentiate at t = 0, or use the mean and variance formulas as a check.
7. 7State the domain of t where the MGF is valid if the question asks for it, and give a clear final answer.

Quickest way: Product rule and PGF substitution

Common mistakes in Sums of Random Variables and Compound Distributions

• Multiplying MGFs when the variables are not independent.

  The product rule is memorised without its condition.

  Fix: Write 'by independence' before every product. If the variables are dependent, the product rule does not hold in general.

• Writing the compound MGF as M_N(M_X(t)).

  Mixing up the PGF and the MGF of N.

  Fix: The correct form is G_N(M_X(t)) = M_N(ln M_X(t)). Check by setting t = 0: M_X(0) = 1, so M_S(0) = G_N(1) = 1.

• Treating the sum of n iid variables as n times one variable.

  Confusing nX with X₁ + … + Xₙ.

  Fix: The MGF of nX is M_X(nt). The MGF of the sum of n iid copies is [M_X(t)]ⁿ. They have different variances.

• Using Var(S) = E[N]Var(X) and forgetting the second term.

  The variance of N is overlooked when N is random.

  Fix: Use Var(S) = E[N]Var(X) + Var(N)(E[X])². Both sources of randomness count.

• Adding means of distributions with different rates and claiming a gamma result.

  Over-generalising the reproductive property of gammas.

  Fix: Independent gammas add their shape parameters only if the rate is the same. Check the MGF form (λ/(λ − t))^α.

• Ignoring the case N = 0 when working from first principles.

  The sum is assumed to have at least one term.

  Fix: Include P(N = 0). In the MGF approach it is handled automatically because the PGF includes the s⁰ term.

Worked examples