Markov’s inequality

For any real random variable $𝑋 : 𝜉 \to ℝ$ and $𝑎 \geq 0$ we have prob

ℙ (| 𝑋 | \geq 𝑎) \leq 𝔼 [| 𝑋 |] 𝑎

Proof

Since $| 𝑋 | \geq 𝑎 1 {| 𝑋 | \geq 𝑎}$ and therefore
$𝔼 [| 𝑋 |] \geq 𝑎 𝔼 [1 {| 𝑋 | \geq 𝑎}] = 𝑎 ℙ (| 𝑋 | \geq 𝑎)$
as required.