Probability 3 - Inequalities

Third post in the series. 

Here is the summary of various inequalities:

Jensen's inequality

It relates the value of a convex function of an integral to the integral of the convex function. if $X$ is a random variable and $\phi$ is a convex function, then

$$\phi(\mathbb{E}[X])\leq\mathbb{E}[\phi(X)]$$ .

Markov's inequality

Let $X$ be a random variable and $a>0$

$$P(|X| \geq a) \leq \frac{\mathbb{E}[|X|]}{a}$$

Chebyshev’s inequality

Let $X$ be a random variable with finite expected value \mu and finite non zero variance \sigma^2. Then for any real number $k>0$

$$P(|X-\mu| \geq k\sigma) \leq \frac{1}{k^2}$$

or written differently: $$P(|X-\mu| \geq k) \leq \frac{Var(X)}{k^2}$$

Chebyshev's inequality follows from Markov's inequality by considering the random variable $(X-\mathbb{E}[X])^2$

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