Tuesday, September 4, 2012

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


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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