proof of Chebyshev’s inequality
The proof of Chebyshev’s inequality follows from the application of Markov’s inequality.
Define Y=(X-μ)2. Then Y≥0 is a random variable, and
𝔼[Y]=Var[X]=σ2. |
Applying Markov’s inequality to Y, we see that
ℙ{|X-μ|≥t}=ℙ{Y≥t2}≤1t2𝔼[Y]=σ2t2. |
Title | proof of Chebyshev’s inequality |
---|---|
Canonical name | ProofOfChebyshevsInequality |
Date of creation | 2013-03-22 12:47:58 |
Last modified on | 2013-03-22 12:47:58 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 6 |
Author | PrimeFan (13766) |
Entry type | Proof |
Classification | msc 60A99 |