# Questions tagged [probability-inequalities]

Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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### Does a sample version of the one-sided Chebyshev inequality exist?

I am interested in the following one-sided Cantelli's version of the Chebyshev inequality: $$\mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,.$$ Basically, ...
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### Random variables for which Markov, Chebyshev inequalities are tight

I am interested in constructing random variables for which Markov or Chebyshev inequalities are tight. A trivial example is the following random variable. $P(X=1)=P(X=-1) = 0.5$. Its mean is zero, ...
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### Understanding proof of a lemma used in Hoeffding inequality

I am studying Larry Wasserman's lecture notes on Statistics which uses Casella and Berger as its primary text. I am working through his lecture notes set 2 and got stuck in the derivation of lemma ...
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