Statistics Solutions

(All Steps) (i) Suppose E X=0 and var;(X)=σ^2 P(X ≥q a) ≤q (σ^2)/(σ^2+a^2), a>0 (ii) Suppose X ≥q 0 and E X^2 P(X>0) ≥q ((E

Question: (i) Suppose \(E X=0\) and \(\operatorname{var}(X)=\sigma^{2}<\infty\). Prove

\[P(X \geq a) \leq \frac{\sigma^{2}}{\sigma^{2}+a^{2}}, a>0\]

(ii) Suppose \(X \geq 0\) and \(E X^{2}<\infty\). Prove

\[P(X>0) \geq \frac{(E X)^{2}}{E X^{2}}\]

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