 Solution

ID: past-exam-of-the-graph-theory-and-probabilistic-combinatorics-course-of-the-university-of-oxford/2009/b/4/a/i/solution

X

has finite mean

μ

σ^{2}

, then for

t > 0

P (∣ X - μ ∣ \geq t) \leq \frac{σ ^{2}}{t ^{2}} .

Indeed

(X - μ)^{2} \geq t^{2} 1_{{∣ X - μ ∣ \geq t}}

pointwise. Taking expectations proves the result, including the zero-variance case.

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