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IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables

X and Y are independent, with X ~ Exponential with mean 2 and Y ~ Exponential with mean 4. What is P(X > Y)?

P(X > Y) = 1/3. With rates 1/2 for X and 1/4 for Y, the chance that X exceeds Y is the rate of Y divided by the sum of the rates, which is (1/4)/(3/4).

  1. A1/3Correct
  2. B1/2
  3. C2/3
  4. D1/4
  5. 3/4

Explanation

Rates are λ_X=1/2 and λ_Y=1/4. P(X>Y)=λ_Y/(λ_X+λ_Y)=(1/4)/(3/4)=1/3. Check by integration: ∫(1/4)e^{-y/4}e^{-y/2}dy=(1/4)/(3/4)=1/3. The value 2/3 is P(X<Y), which wrongly uses the rate of X in the numerator.

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