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IAI Actuarial Core Principles · Actuarial Statistics · Generating functions

Claims counts N follow a Poisson distribution with mean 4. The cumulant generating function is K(t) = 4(e^t - 1). What is the third central moment of N, and the skewness?

Every cumulant of a Poisson(4) variable equals 4, so the third central moment is 4. Skewness is the third cumulant divided by variance to the power 1.5, which is 4/8 = 0.5.

  1. AThird central moment 4; skewness 1
  2. BThird central moment 4; skewness 0.5Correct
  3. CThird central moment 12; skewness 1.5
  4. DThird central moment 64; skewness 0.5
  5. Third central moment 4; skewness 2

Explanation

K'''(t) = 4e^t, so the third cumulant at 0 is 4, which equals the third central moment. The variance is 4, so the skewness is 4/4^1.5 = 4/8 = 0.5. Option 1 wrongly divides by the variance instead of the variance^1.5.

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