IAI Actuarial Core Principles · Actuarial Statistics · Generating functions

X has an exponential distribution with MGF M(t) = 4/(4 - t) for t < 4. What is P(X > 0.5)?

The MGF identifies an exponential with rate 4. The survival probability is e^(-4 × 0.5) = e^-2. The option 1 - e^-2 gives the lower tail, not P(X > 0.5).

Ae^-4
Be^-2Correct
C1 - e^-2
De^-0.125
0.5

Explanation

M(t) = λ/(λ - t) with λ = 4, so X ~ Exp(4). P(X > 0.5) = e^(-4×0.5) = e^-2. Option 1 - e^-2 is P(X ≤ 0.5), the wrong tail.

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