Paper 2, Section II, F

Probability | Part IA, 2015

State and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.

If XX is a random variable with mean zero and finite variance σ2\sigma^{2}, prove that for any a>0a>0,

P(Xa)σ2σ2+a2P(X \geqslant a) \leqslant \frac{\sigma^{2}}{\sigma^{2}+a^{2}}

[Hint: Show first that P(Xa)P((X+b)2(a+b)2)P(X \geqslant a) \leqslant P\left((X+b)^{2} \geqslant(a+b)^{2}\right) for every b>0b>0.]

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