 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/probability-and-computing/1/c/solution

X

with finite expectation

μ

and finite variance

σ^{2}

. For every real

a > 0

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

No independence assumption is needed. Applying Markov's inequality to the nonnegative random variable

(X - μ)^{2}

proves the formula, including the zero-variance case.

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