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Probability: Difference between revisions
→Chebyshev's Inequality
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===Markov's Inequality=== | ===Markov's Inequality=== | ||
===Chebyshev's Inequality=== | ===Chebyshev's Inequality=== | ||
* <math>P(|X - \mu| \geq k \sigma) \leq \frac{1}{k^2}</math> | |||
* <math>P(|X - \mu| \geq k) \leq \frac{\sigma^2}{k^2}</math> | |||
{{hidden | Proof | | |||
Apply Markov's inequality:<br> | |||
Let <math>Y = |X - \mu|</math> | |||
<math>P(|X - \mu| \geq k) = P(Y \geq k) = = P(Y^2 \geq k^2) \leq \frac{E(Y^2)}{k^2} = \frac{E((X - \mu)^2)}{k^2}</math> | |||
}} | |||
* Usually used to prove convergence in probability | |||
===Central Limit Theorem=== | ===Central Limit Theorem=== | ||
Very very important. Never forget this.<br> | Very very important. Never forget this.<br> | ||