 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-12/1/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Let

Z

m

and finite variance

σ^{2}

. The Chebyshev inequality is

P (∣ Z - m ∣ \geq t) \leq \frac{σ ^{2}}{t ^{2}} (t > 0) .

The proof is the pointwise comparison

t^{2} 1_{{∣ Z - m ∣ \geq t}} \leq (Z - m)^{2} .

Taking expectations and using the definition of variance gives

t^{2} P (∣ Z - m ∣ \geq t) \leq E (Z - m)^{2} = σ^{2} .

Division by the positive number

t^{2}

proves the Chebyshev inequality, including the case

σ^{2} = 0

. Equivalently, this is the Markov inequality applied to the nonnegative random variable

(Z - m)^{2}

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