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Probability: Difference between revisions
→Chebyshev's Inequality
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Apply Markov's inequality:<br> | Apply Markov's inequality:<br> | ||
Let <math>Y = |X - \mu|</math><br> | Let <math>Y = |X - \mu|</math><br> | ||
Then <math>P(|X - \mu| \geq k) = P(Y \geq k) | Then:<br> | ||
<math> | |||
\begin{aligned} | |||
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} | |||
\end{aligned} | |||
</math> | |||
}} | }} | ||
* Usually used to prove convergence in probability | * Usually used to prove convergence in probability | ||