NAVIGATION
Home
Research
Bookshelf
Garden
FIND ME ON
GitHub
LinkedIn
Email
Theorem (Chebyshev Inequality)
For X∈L2X\in \mathscr{L}^{2}X∈L2: P(∣X−μ∣≥α)≤Var(X)α2\mathbb{P}(|X-\mu|\ge \alpha)\le \frac{\text{Var}(X)}{\alpha^{2}}P(∣X−μ∣≥α)≤α2Var(X)for all α≥0\alpha\ge 0α≥0.
\begin{proof} P(∣X−μ∣≥α)=P(∣X−μ∣2≥α2)≤E[∣X−μ∣2]α2\mathbb{P}(|X-\mu|\ge\alpha)=\mathbb{P}(|X-\mu|^{2}\ge\alpha^{2})\le \frac{\mathbb{E}[|X-\mu|^{2}]}{\alpha^{2}}P(∣X−μ∣≥α)=P(∣X−μ∣2≥α2)≤α2E[∣X−μ∣2] with the inequality being an application of Markov’s Inequality. \end{proof}
\begin{proof}
\end{proof}
Law of Large Numbers
Summary of MATH 895