 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-2/3f/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The Markov inequality states that for a nonnegative random variable

Y

and

a > 0

Pr (Y \geq a) \leq \frac{E Y}{a} .

For

t > 0

X \geq x

is the same as

e^{tX} \geq e^{t x}

. Applying Markov's inequality to the nonnegative variable

e^{tX}

Pr (X \geq x) \leq e^{- t x} M_{X} (t) .

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