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IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions

Ten independent claim sizes are drawn from a continuous distribution, and the 10 values are ordered X(1) < ... < X(10). Let m be the population median. What is P(X(3) < m < X(8))? Use the fact that the number of observations below m is Binomial(10, 0.5). Choose the nearest value.

The probability is about 0.891. The interval holds the median when between 3 and 7 of the 10 observations fall below it. The binomial counts 120+210+252+210+120 total 912 out of 1024.

  1. A0.891Correct
  2. B0.945
  3. C0.656
  4. D0.980
  5. 0.773

Explanation

X(3) < m < X(8) requires at least 3 and at most 7 observations below m, i.e. N in {3,...,7}. Binomial(10,0.5) counts out of 1024 are: 120, 210, 252, 210, 120 for N = 3 to 7, summing to 912. 912/1024 = 0.8906. Including N = 2 or 8 would give about 0.98, which is wrong.

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