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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

A non-negative integer-valued random variable N has P(N > n) = 0.6^n for n = 0, 1, 2, .... What is E[N]?

Using E[N] = Σ P(N > n) for n ≥ 0, the sum is the geometric series Σ0.6^n = 1/0.4 = 2.5. This matches a geometric distribution on 1,2,3,... with success probability 0.4.

  1. A1.50Correct
  2. B0.60
  3. C2.50
  4. D1.00
  5. 1.67

Explanation

For non-negative integer variables, E[N] = sum over n≥0 of P(N > n) = sum of 0.6^n = 1/(1-0.6) = 2.5. Checking: P(N=0) = 1 - 0.6 = 0.4, so N is geometric on 1,2,... with p=0.4, giving mean 1/0.4 = 2.5. Hence the key is 2.5 and not 1.5.

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