Acad | Proof of useful inequalities in Econometrics
date
Aug 14, 2025
slug
prof-ineq-econometric
status
Published
summary
Markov’s Inequality, Chebyshev’s Inequality, and Cauchy—Schwarz Inequality
tags
Academic
Data Analysis
Business
Finance
Math
Economics
type
Post
Markov’s InequalityProofExtensionChebyshev’s InequalityProofApplicationCauchy-Schwarz inequalityProofApplication & other forms
Markov’s Inequality
If is a non-negative random variable, for any
Proof
Extension
Chebyshev’s Inequality
The Chebyshev’s Inequality is closely related to Markov’s Inequality, by replacing the random variable with . However, Chebyshev’s inequality does not require :
Proof
Application
In Normal Distribution,
Cauchy-Schwarz inequality
The Cauchy–Schwarz inequality is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms.
Proof
Application & other forms
In , Cauchy Inequaliy:
In probability:
where equality holds if and only if for some constant