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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling

For a compound Poisson variable S with Poisson parameter λ and claim size X, which expression gives the third central moment of S?

The third central moment of a compound Poisson variable is λE[X³]. Its cumulant generating function is λ(M_X(t) − 1), so each cumulant of order r equals λE[X^r], and the third cumulant equals the third central moment.

  1. AλE[X³]Correct
  2. BλE[X²]
  3. Cλ(E[X])³
  4. DE[X³] − λ
  5. λ²E[X³]

Explanation

The cumulant generating function of S is λ(M_X(t) − 1), so the r-th cumulant is λE[X^r]. The third cumulant equals the third central moment, giving λE[X³]. Using λE[X²] gives the variance instead.

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