IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X and Y are independent exponential random variables with rates 1 and 2 respectively. What is P(X < Y)?
The probability is 1/3. For independent exponentials, P(X < Y) equals the rate of X divided by the sum of rates, 1/(1+2) = 1/3. The variable with the larger rate tends to be smaller, so X is less likely to be the smaller one.
- A1/2
- B2/3
- C1/3Correct
- D1/4
- 3/4
Explanation
For independent exponentials, P(X < Y) = λX/(λX + λY) = 1/(1+2) = 1/3. Option 2/3 is P(Y < X), found by reversing the rates.
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