Markov’s Inequality

Statement

If 𝑋 is a nonnegative random variable and 𝑎>0, then the probability that 𝑋 is at least 𝑎 is at most the expectation of 𝑋 divided by 𝑎:

• 𝐏⁡(𝑋≥𝑎) ≤ 𝐄[𝑋]/𝑎

Let 𝑎 = 𝑎̃⋅𝐄[𝑋] where 𝑎̃>0; then we can rewrite the previous inequality as

• 𝐏⁡(𝑋 ≥ 𝑎̃⋅𝐄[𝑋]) ≤ 1/𝑎̃

In the language of measure theory, Markov’s inequality states that if:

• (𝑋, 𝛴, 𝜇) is a measure space
• 𝑓 is a measurable extended real-valued function
• 𝜀>0

then:

• $𝜇(x\in X:|f(x)|\ge 𝜀)\le \frac{1}{𝜀}{\int }_X|f|d𝜇$

This measure-theoretic definition is sometimes referred to as Chebyshev’s inequality.