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Probability: Difference between revisions
→Chebyshev's Inequality
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{{hidden | Proof | | {{hidden | Proof | | ||
Apply Markov's inequality:<br> | Apply Markov's inequality:<br> | ||
Let <math>Y = |X - \mu|</math> | Let <math>Y = |X - \mu|</math><br> | ||
<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> | Then <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 | * Usually used to prove convergence in probability | ||