Chernoff’s inequality

Let $𝑋 : 𝜉 \to ℝ$ be a real random variable and $𝑎 \geq 0$ and $𝑡 \geq 0$ . Then prob

ℙ (𝑋 \geq 𝑎) \leq 𝔼 [e 𝑡 𝑋] e 𝑡 𝑎

Proof

By Markov’s inequality
$ℙ (e 𝑡 𝑋 \geq e 𝑡 𝑎) \leq 𝔼 [e 𝑡 𝑋] e 𝑡 𝑎$
as required.