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

A random variable X takes the values 0, 1, 2 and 4 with probabilities 0.1, 0.3, 0.4 and 0.2 respectively. What is E[X]?

The expectation is the probability-weighted sum of the values: 0×0.1 + 1×0.3 + 2×0.4 + 4×0.2 = 1.9. Each value must be multiplied by its own probability, not averaged equally, so the correct answer is 1.9.

  1. A1.5
  2. B1.9Correct
  3. C2.0
  4. D1.7
  5. 2.3

Explanation

E[X] = 0(0.1) + 1(0.3) + 2(0.4) + 4(0.2) = 0 + 0.3 + 0.8 + 0.8 = 1.9. The value 1.75 style errors arise from averaging the values equally: (0+1+2+4)/4 = 1.75, which is not an option here, while 1.5 comes from ignoring the weight on 4.

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