# Proof of Chebyshev's inequality question

In this proof of the Chebyshev's inequality, why is it assumed that $$f_{X-E(X)}=f_{(X-E(X))^2}$$, where$$f$$ is the probability density function?

The second line in this proof says:

$$\operatorname {Var} (X)=\sigma ^{2}=\int _{\mathbb {R} }(x-\mu )^{2}f(x)\,dx,$$

shouldn't the p.d.f. be $$f_{(X-E(X))^2}(x)$$ instead of $$f(x)$$? In the proof later, they use:

$$\Pr(|X-\mu |\geq k\sigma )=\int _{|x-\mu |\geq k\sigma }f(x)\ dx,$$ which means that their $$f(x)$$ equals by definition to $$f_{X-E(X)}(x)$$. So, again, why is it assumed that $$f_{X-E(X)}=f_{(X-E(X))^2}$$?

• The Wikipedia proof never mentions "$f_{X-E(X)}(x)$" or any such monstrosities... – Lord Shark the Unknown Dec 17 '19 at 20:27
• @lord-shark-the-unknown yes but they write down p.d functions of both random variables $X-E(X)$ and $(X-E(X)^2$ under the same name $f$ and i dont understand how they use them interchangeably edit: $f$ and not $f(X)$, sorry – Nick The Dick Dec 17 '19 at 20:30
• $f(x)$ is the PDF of $X$. The line $Var(X)$ is how the variance of the r.v. $X$ is $defined$, the $f$ inside that integral is still the PDF of $X$. Perhaps you should review the definitions of PDF and variance to clear up your confusion, or add more content to your question to guide the answers to more specific grounds. – Fede Poncio Dec 17 '19 at 20:40
• @FedePoncio from WIkipedia: "If $X$ is a random variable whose cumulative distribution function admits a density $f(x)$, then the expected value is defined as the following Lebesgue integral, if the integral exists: $\operatorname {E} [X]=\int _{\mathbb {R} }xf(x)\,dx.$" So, for r.v. $X-E(X)$ there should be a different PDF? – Nick The Dick Dec 17 '19 at 20:47
• In your notation, $$E[\phi(x)]=\int_{\Bbb R}\phi(x)f(x)\,dx$$ for any reasonable function $\phi$. – Lord Shark the Unknown Dec 17 '19 at 20:55