I am confused by the following (possibly simple) algebra in the proof of the Cantelli-Cheybyshev inequality. I am following Rohatgi and Saleh (2015, Section 3.4 Lemma 1) where we plug in $\phi(x)=(x+c)^2$ into Markov's inequality to get:
\begin{equation} P(Y \geq t) \leq P\big( \phi(Y) \geq (t+c)^2 \big) \leq \underbrace{\frac{{E}[(Y+c)^2]}{(t+c)^2}}_{RHS} \end{equation} where $EY=0, t\geq 1,$ and $c>0$. To find the tightest bound, we minimize the RHS wrt to $c$. This gives us \begin{equation} c^* = \frac{Var Y}{t} \end{equation} The proof in Rohatgi and Saleh (2015) and elsewhere (e.g. Cantelli's inequality proof) says plugging in $c^*$ gives the desired expression: \begin{equation} P(Y \geq t) \leq \frac{Var Y}{Var Y + t^2} \end{equation} But the algebra in between doesn't seem trivial (at least to me): if I plug in $c^*$ into RHS, I don't see how to get the RHS of Cantelli's inequality even knowing that I can add or substract $EY=0$: \begin{equation} \frac{{E}[(Y+c)^2]}{(t+c)^2}\bigg|_{c=c^*} = \frac{Var Y + (Var Y/t)^2}{t^2 + 2t(Var Y/t) + (Var Y/t)^2} = ...? \end{equation}

Any help would be greatly appreciated!


1 Answer 1


Hint. Note that, factorizing the polynomial:

$$t^2 + 2t(Var Y/t) + (Var Y/t)^2=(t+Var Y/t)^2=\frac{1}{t²}(t²+Var Y)^2$$


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