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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

The sum S of 50 independent Poisson(2) claim counts is approximated by a normal distribution with continuity correction. For which statement about P(S = 100) is the approximation correct?

P(S = 100) is approximated by P(99.5 < Y < 100.5) where Y is normal with mean 100 and variance 100, since S is Poisson(100) and each integer maps to a unit-width interval.

  1. AIt is P(99.5 < Y < 100.5) with Y ~ N(100, 100)Correct
  2. BIt is P(99 < Y < 101) with Y ~ N(100, 100)
  3. CIt is P(99.5 < Y < 100.5) with Y ~ N(100, 10)
  4. DIt is zero because Y is continuous
  5. It is P(99.5 < Y < 100.5) with Y ~ N(100, 50)

Explanation

S is Poisson(100), so mean 100 and variance 100 (standard deviation 10). A single integer value is approximated by the interval from 99.5 to 100.5. Option N(100,10) confuses variance with standard deviation, and answering zero ignores the correction.

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