Difference between revisions of "Delta Tschebycheff inequality"
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| + | ==Theorem== | ||
Let $\mathbb{T}$ be a [[time scale]] and let $\epsilon > 0$. Then | Let $\mathbb{T}$ be a [[time scale]] and let $\epsilon > 0$. Then | ||
$$\dfrac{\mathbb{V}ar_{\mathbb{T}}(X) - \mathbb{E}_{\mathbb{T}}(2H(X))}{\epsilon^2} \geq P((X-\mathbb{E}_{\mathbb{T}}(X))^2 \geq \epsilon^2),$$ | $$\dfrac{\mathbb{V}ar_{\mathbb{T}}(X) - \mathbb{E}_{\mathbb{T}}(2H(X))}{\epsilon^2} \geq P((X-\mathbb{E}_{\mathbb{T}}(X))^2 \geq \epsilon^2),$$ | ||
where the density function of $H(X)$ is $h_2(t,0)-\dfrac{t^2}{2}$. | where the density function of $H(X)$ is $h_2(t,0)-\dfrac{t^2}{2}$. | ||
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| + | ==Proof== | ||
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| + | ==References== | ||
{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:38, 15 September 2016
Theorem
Let $\mathbb{T}$ be a time scale and let $\epsilon > 0$. Then $$\dfrac{\mathbb{V}ar_{\mathbb{T}}(X) - \mathbb{E}_{\mathbb{T}}(2H(X))}{\epsilon^2} \geq P((X-\mathbb{E}_{\mathbb{T}}(X))^2 \geq \epsilon^2),$$ where the density function of $H(X)$ is $h_2(t,0)-\dfrac{t^2}{2}$.
Proof
References
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
