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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 3F / c / Solution

For any

t > 0

, the exponential Markov bound and part (b) give

P (X \geq v) \leq e^{- t v} M_{X} (t) = e^{- t v + t^{2} /2} .

The minimum occurs at

t = v

, so the Gaussian tail bound is

\int_{v}^{\infty} \frac{e ^{- x^{2} /2}}{2 π} d x \leq e^{- v^{2} /2} .

Solved by gpt-5.6-sol high.

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