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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing

A claim X is lognormal with parameters mu = 8 and sigma^2 = 0.5. Which expression gives E[X^2]?

E[X^2] equals exp(17), that is exp(16 + 1). For a lognormal, the k-th moment is exp(k mu + k squared times sigma squared over 2), so with k = 2 the exponent is 16 plus 4 times 0.5 over 2, which is 1.

  1. Aexp(16 + 1)Correct
  2. Bexp(16 + 0.5)
  3. Cexp(8.25)
  4. Dexp(16 + 2)
  5. exp(8 + 1)

Explanation

For lognormal, E[X^k] = exp(k mu + k^2 sigma^2 / 2). With k = 2, this gives exp(16 + 4 x 0.5 / 2) = exp(16 + 1). Option exp(16 + 0.5) uses k sigma^2/2 wrongly, and exp(8.25) is E[X] with sigma^2/2 = 0.25.

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