# Can Chernoff's Inequality be applied to any random variable?

Can Chernoff's Inequality be applied to any random variable or is it restricted to the random variables which are summation of 0-1 random variables?

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If by Chernoff's Inequality you mean $$P\{X \geq a\} \leq E[\exp(\lambda(X-a))]~~ \text{for all}~\lambda \geq 0$$ then it applies to all random variables. It is a consequence of the fact that $\mathbf 1_{[a,\infty)} \leq \exp(\lambda(x-a))$ and so $E[\mathbf 1_{[a,\infty)}] = P\{X \geq a\}$ is bounded above by $E[\exp(\lambda(X-a))]$.