# Notes - Probability MT22, Chebyshev's inequality

> Source: https://ollybritton.com/notes/uni/prelims/mt22/probability/notes/chebyshevs-inequality/ · Updated: 2025-10-21 · Tags: uni, notes

- [Course - Probability](https://ollybritton.com/notes/uni/prelims/mt22/probability/)

### Flashcards
@State Chebyshev’s inequality for a random variable $Z$ with finite variance.::

$$
\forall c>0, \quad \mathbb{P}(|Z - \mathbb{E}[Z]| \ge c) \le \frac{\mathrm{Var}(Z)}{c^2}
$$

@Prove Chebyshev’s inequality (∆chebyshevs-inequality): for a random variable $X$ with mean $\mu$ and variance $\sigma^2$, and any $t > 0$,
$$
\mathbb{P}(|X-\mu| \ge t) \le \frac{\sigma^2}{t^2}
$$

::

@todo?

@important~

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