Prove that if \(\mathbb {P}\{X\geq 0\}=1\) and \(0<\mathbb {E}(X^2)<\infty \), then \[ \mathbb {P}\{X=0\}\leq \frac {\operatorname {Var} X}{\mathbb {E}(X^2)}. \]
Question
Math 6010 • Linear Models
Fall 2009 • Question 9
Question
Fall 2009 • Question 9
Prove that if \(\mathbb {P}\{X\geq 0\}=1\) and \(0<\mathbb {E}(X^2)<\infty \), then \[ \mathbb {P}\{X=0\}\leq \frac {\operatorname {Var} X}{\mathbb {E}(X^2)}. \]