202405302108
Status: #idea
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State: #nascient

Chebyshev's Inequality

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Chebyshev's Inequality really is just a special case of Markov's Inequality where we replace $X$ with $X - b$ :

P (| X - b | \geq a) \leq \frac{E (| X - b |)}{a}

Equivalently we can do the following:

P [(X - b)^{2} \geq a^{2}] \leq \frac{E [(X - b)^{2}]}{a^{2}}

Observe that $(X - b)^{2} \geq a^{2}$ exactly when $| X - b | \geq a$ , so they are really equivalent statements!

Now how do we put that in the format we're familiar with?
Simply pick $b = μ$ and $a = k σ$ such that:

\begin{aligned} P (| X - μ | & \geq k σ) \leq \frac{E [(X - μ)^{2}]}{k^{2} σ^{2}}, we write the LHS and RHS in different but equivalent manners \\ \leq \frac{σ^{2}}{k^{2} σ^{2}}, we can replace numerator by Var(X) \\ \leq \frac{1}{k^{2}} \end{aligned}

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