A bound for a sample proportion

Topic: Expectation and convergence · more questions on this topic
‹ A spike that keeps its mean

Let \overline Y_n be the sample proportion in n independent Bernoulli(p) trials. Use Chebyshev’s inequality to bound \mathrm{P}(|\overline Y_n-p|>\varepsilon).

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A Bernoulli(p) variable has variance p(1-p), so \overline Y_n has mean p and variance p(1-p)/n. Chebyshev’s inequality gives

\mathrm{P}(|\overline Y_n-p|>\varepsilon)\le\frac{p(1-p)}{n\varepsilon^2}.