arrow_back Prove Chebyschev’s inequality

28 views
Prove Chebyschev&rsquo;s inequality

(Chebyschev's inequality). Let $X$ be a random variable. Then
$P(|X-E X|>\epsilon) \leq \frac{\operatorname{Var}(X)}{\epsilon^{2}}$

Proof.

This is marked as a corollary because we simply apply Markov's inequality to the nonnegative random variable $(X-E X)^{2}$. We then have
$\begin{array}{rlr}P(|X-E X|>\epsilon) & =P\left((X-E X)^{2}>\epsilon^{2}\right) & (\text { statements are equivalent }) \\ & \leq \frac{E\left[(X-E X)^{2}\right]}{e^{2}} & \text { (Markov's inequality) } \\ & =\frac{\operatorname{Var}(X)}{\epsilon^{2}} & \text { (definition of variance) }\end{array}$
(definition of variance)
by Platinum
(106,844 points)

Related questions

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove Markov’s inequality.
Prove Markov’s inequality.Prove (Markov's inequality). Suppose $X$ is a nonnegative random variable and $a \in$ is a positive constant. Then \ P(X \geq a) \leq \frac{E X} ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove the Prohorov inequality
Prove the Prohorov inequalityProve the Prohorov inequality ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove Kolmogorov's inequality
Prove Kolmogorov's inequalityProve Kolmogorov's inequality ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
State Kolmogorov's inequality
State Kolmogorov's inequalityState Kolmogorov's inequality ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove the Bennett inequality
Prove the Bennett inequalityProve the Bennett inequality ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $a, b, c$ be the side lengths, and $h_a, h_b, h_c$ be the altitudes, respectively, and $r$ be the inradius of a triangle. Prove the inequality ....
Let $a, b, c$ be the side lengths, and $h_a, h_b, h_c$ be the altitudes, respectively, and $r$ be the inradius of a triangle. Prove the inequality ....Let $a, b, c$ be the side lengths, and $h_a, h_b, h_c$ be the altitudes, respectively, and $r$ be the inradius &nbsp;of a triangle. Prove the ...
Prove that, for all values of $x$, $x^{2}+4 x+12>x+3$
Prove that, for all values of $x$, $x^{2}+4 x+12>x+3$Prove that, for all values of $x$, \ x^{2}+4 x+12&gt;x+3 \ ...