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

Let X be uniform on (0, 10) and let Y = min(X, 4), representing a claim capped at Rs 4 lakh. What is E[Y]?

Capping at 4 gives E[Y] = ∫0^4 x(0.1)dx + 4×P(X>4) = 0.8 + 2.4 = 3.2. The same result follows from integrating the survival function from 0 to 4, giving 4 − 0.8 = 3.2.

  1. A2.8
  2. B3.2
  3. C3.6Correct
  4. D4.0
  5. 2.0

Explanation

E[Y] = integral from 0 to 4 of x(0.1) dx + 4·P(X>4) = 0.1·8 + 4·0.6 = 0.8 + 2.4 = 3.2. Alternatively E[Y] = integral from 0 to 4 of P(X>x) dx = integral of (1 - x/10) = 4 - 0.8 = 3.2. So the key is 3.2; 3.6 would come from wrongly using P(X>4)=0.6 with weight 6 or forgetting the integral part.

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