Markov’s inequality
Topic of Stochastic processes, part of Stochastic processes · Also in Functions of random variables and limit theorems
Markov’s inequality
| Type | Topic |
| Taught in | Stochastic processes · Functions of random variables and limit theorems |
| Part | Stochastic processes · Probability |
| Questions using it | 1 |
For a non-negative random variable X and a>0, \mathbb{P}(X\ge a)\le\mathbb{E}X/a. It needs nothing but the mean, and applied to (X-\mathbb{E}X)^2 it gives Chebyshev’s inequality.
In Stochastic processes
From Stochastic processes
For a non-negative random variable and a>0, Markov’s inequality is the comparison X\ge a\,1_{\{X\ge a\}}. Taking expectations gives a\mathbb{P}(X\ge a)\le\mathbb{E} X, so
\mathbb{P}(X\ge a)\le\frac{\mathbb{E} X}{a}.
In Functions of random variables and limit theorems
Chebyshev’s inequality, used above, is the general bound \mathbb{P}(|X-\mathbb{E} X|\ge \varepsilon)\le \operatorname{Var}(X)/\varepsilon^2, and Markov’s inequality is \mathbb{P}(X\ge x)\le \mathbb{E} X/x for a non-negative X and x>0.
Used in
- Taught in Stochastic processes · Functions of random variables and limit theorems
- Also used in Functions of random variables and limit theorems · Galton–Watson branching · Expectation and convergence
- Questions that use it (1) — L2 convergence and probability
- All 2 questions on Stochastic processes
See also
Chebyshev’s inequality · Jensen’s inequality · Cauchy–Schwarz inequality · Measurable random variable · Stochastic process · Stationary and independent increments · What a stochastic process models · State, clock, and assumptions