Introduction to Probability · Problem 8.1.6

Law of Large Numbers — Grinstead & Snell

Mathematics — Probability · chebyshev-inequality-bernoulli

Problem

Let SnS_n be the number of successes in nn Bernoulli trials with probability pp for success on each trial. Show, using Chebyshev's Inequality, that for any ϵ>0\epsilon > 0 P(∣Snn−p∣≥ϵ)≤p(1−p)nϵ2.P\left(\left|\frac{S_n}{n} - p\right| \geq \epsilon\right) \leq \frac{p(1-p)}{n\epsilon^2}.

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