Inequalities

Title

Context:

• Source:
• Relation:

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Insight:

• Markov’s Inequality
  • X is RV, nonnegative, for any a > 0 $P(X\ge a)\le \frac{E[X]}{a}$
• Chebyshev’s Inequality
  • X is RV, finite mean $\mu$, variance ${\sigma }^2$, for any k > 0 $P(|X-\mu |\ge k)\le \frac{{\sigma }^2}{k^2}$
  • One-side Chebyshev
    • If $E[X]=\mu$ and Var(X) = ${\sigma }^2$, for a > 0 $\begin{array}{c}P\{X\ge \mu +a\}\le \frac{{\sigma }^2}{{\sigma }^2+a^2}\\ P\{X\le \mu -a\}\le \frac{{\sigma }^2}{{\sigma }^2+a^2}\end{array}$
• Purpose of Inequalities above
  • derive bounds on probabilities when mean (and variance) are known
• If $Var(X)=0$, then $P(X=E[X])=1$: the only random variables with variance = 0 are those constant + with probability of 1