Important Theorems and Inequalities

Triangle Inequality

Cauchy-Schwarz Inequality

Linear Algebra

Probability

Markov’s Inequality

Chebyshev’s Inequality

Chernoff’s Inequality

Central Limit Theorem

Jensen’s Inequality

For a convex function $f$ and a random variable $X$:

$$f(\mathbb{E}[X])\le \mathbb{E}[f(X)]$$

Similarly, for a concave function $g$ and a random variable $Y$, the inequality says:

$$g(\mathbb{E}[Y])\ge \mathbb{E}[g(Y)]$$

Chain Rule of Probability