Chebyshev's Inequality Proof Step

Question

I'm trying to understand a (maybe straightforward) step in the proof of Chebyshev's inequality:

$$ P(|X-\bar X| \geq a) \leq \frac{\sigma^{2}}{a^{2}} \quad (1) $$

The proof starts with the Markov's inequality $$ P(X \geq a) \leq \frac{E[X]}{a} \quad (2) $$

And it is used to prove Chebyshev's inequality by replacing $X$ by $Y = (X - \bar X)^2$, which results in

$$ P((X-\bar X)^2 \geq a^2) \leq \frac{E[(X-\bar X)^2]}{a^{2}} \quad (3) $$

Then it gives (1)

Why is it $a^2$ in the right-hand side $\frac{E[(X-\bar X)^2]}{a^{2}}$ and just not $a$?

Answer 1

You actually use the variable

$$ |X-E(X)|$$

So putting square root in the probability doesn't affected the right term.

Answer 2

I think changing variables name can be helpful. Markov inequality states for every nonnegative random variable $Z$ and every positive $b$ that: \begin{equation*} P(Z \geq b) \leq \frac{E[Z]}{b}. \end{equation*} You need to apply this inequality with $Z=(X-\bar{X})^2$ and $b=a^2$. Keep in mind that: \begin{equation*} (X-\bar{X})^2 \geq a^2 \Leftrightarrow \lvert X-\bar{X} \rvert \geq a \end{equation*} since $a$ is positive.

Answer 3

That is a good question. It is a small point that has to do with a change of variables. Markov's Inequality says that for any $a \in \mathbb{R}_{>0},\; P(X \geq a) \leq \frac{\mathbb{E}[X]}{a}$. Therefore, we can let $b = a^2$. For clarity, let's prove Chebyschev's Inequality outright from Markov's $$ \begin{align} P(X \geq b) \leq \frac{\mathbb{E}[X]}{b}\\ \textit{Let } Y = (X-\overline{X})^2 \\ \Rightarrow P((X-\overline{X})^2 \geq b) \leq \frac{\mathbb{E}[(X-\overline{X})^2]}{b} \\ \boxed{\text{Let } b = a^2} \\ \Rightarrow P((X-\overline{X})^2 \geq a^2) \leq \frac{\mathbb{E}[(X-\overline{X})^2]}{a^2} \\ \text{Using } (X-\overline{X})^2 \geq a^2 \Leftrightarrow |(X-\overline{X})| \geq a \\ \qquad \text{and } \mathbb{E}[(X-\overline{X})^2] = Var(X) = \sigma^2 \\ \Rightarrow P(|(X-\overline{X})| \geq a) \leq \frac{\sigma^2}{a^2} \end{align} $$ Let me know if you have any further questions.