 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/18/c/iii/solution

For

t \geq 0

, evaluate the survival function directly:

P (X > t) = λ^{2} \int_{t}^{\infty} x e^{- λ x} d x = (1 + λ t) e^{- λ t} .

t = μ + 2 σ = (2 + 22) / λ

, this becomes

P (X > μ + 2 σ) = (3 + 22) e^{- 2 (1 + 2)} \approx 0.04662.

λ

cancels because the threshold scales with the distribution. Also

μ - 2 σ < 0

lies outside the support, so in this particular case the two-sided event

∣ X - μ ∣ > 2 σ

has the same probability as the stated upper-tail event.

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