Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2010 / B / 4 / a

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- Solution a

Solution

For a nonnegative random variable

Y

and

a > 0

, Markov inequality states

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

For a random variable

X

of finite mean

μ

σ^{2}

, Chebyshev inequality states that for every

t > 0

,

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

To prove it, apply Markov inequality to the nonnegative variable

(X - μ)^{2}

at threshold

t^{2}

:

P (∣ X - μ ∣ \geq t) = P ((X - μ)^{2} \geq t^{2}) \leq \frac{E [( X - μ ) ^{2} ]}{t ^{2}} .

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