Chebyshev’s inequality
Topic of Stochastic processes, part of Stochastic processes · Also in Functions of random variables and limit theorems, Estimation
Chebyshev’s inequality
| Type | Topic |
| Taught in | Stochastic processes · Functions of random variables and limit theorems · Estimation |
| Part | Stochastic processes · Probability · Statistical inference |
| Questions using it | 2 |
For a random variable with finite variance and \varepsilon>0, \mathbb{P}(|X-\mathbb{E}X|\ge\varepsilon)\le\operatorname{Var}(X)/\varepsilon^2. It is Markov’s inequality applied to (X-\mathbb{E}X)^2, and it proves the weak law of large numbers and the consistency of an estimator whose mean squared error tends to zero.
In Stochastic processes
From Stochastic processes
If X has a finite variance, apply the same bound to (X-\mathbb{E} X)^2 at level \varepsilon^2. That is Chebyshev’s inequality:
\mathbb{P}\bigl(|X-\mathbb{E} X|\ge\varepsilon\bigr)\le\frac{\operatorname{Var}(X)}{\varepsilon^2}.
In Functions of random variables and limit theorems
The weak law under a finite variance follows from Chebyshev’s inequality: \mathbb{P}(|S_n/n-\mu|>\varepsilon)\le \operatorname{Var}(S_n/n)/\varepsilon^2=\operatorname{Var}(X_1)/(n\varepsilon^2), which tends to zero.
In Estimation
From Estimation
For a large sample the leading question is whether the estimator settles on \theta. If the mean squared error tends to zero, the estimator is consistent, by Chebyshev’s inequality: \mathbb{P}(|T_n-\theta|>\varepsilon)\le\operatorname{MSE}(T_n)/\varepsilon^2. The law of large numbers says that the sample mean is consistent for \mu, and the same limit passes to continuous functions of a consistent estimator, which is why moment estimators are usually consistent. Both estimators in each example above are consistent. Maximum likelihood estimators are consistent in the regular models used here.
Used in
- Taught in Stochastic processes · Functions of random variables and limit theorems · Estimation
- Also used in Functions of random variables and limit theorems · Expectation and convergence · Estimation
- Questions that use it (2) — A bound for a sample proportion · A Poisson mean (answer)
- All 2 questions on Stochastic processes
See also
Markov’s inequality · Jensen’s inequality · Cauchy–Schwarz inequality · Measurable random variable · Variance · Law of large numbers · Estimator · Mean squared error