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

A random variable N takes the values 0, 1, 2, ... with P(N = n) = (1/2)^(n+1). What is E[N]?

E[N] equals 1. The distribution is geometric on 0, 1, 2, ... with success probability 1/2, so the mean is (1−p)/p = 1. The value 2 would be the mean of a geometric starting at 1.

  1. A0.5
  2. B1Correct
  3. C2
  4. D1.5
  5. Infinite

Explanation

This is a geometric distribution on 0, 1, 2, ... with p = 1/2. Its mean is (1−p)/p = 0.5/0.5 = 1. Check: the sum of n(1/2)^(n+1) = (1/2) times the sum of n(1/2)^n = (1/2)(2) = 1. The answer 2 is the mean of the geometric starting from 1.

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