Inequality

(BOOLE'S INEQUALITY): $P(\bigcup_{k=1}^{\infty} A_k)\le \sum_{k=1} ^{\infty} P(A_k)$

(BONFERRONI'S INEQUALITY): $P(\cap_{k=1}^{\infty} A_k) \ge 1 - \sum_{k=1} ^{\infty} P(A_k^c)$

(MARKOV INEQUALITY): $X$ is a non negative random variable, $\forall a > 0, P(X \ge a)\le \frac{\mathbb{E}[X]}{a}$

(CHEBYCHEV's INEQUALITY): $X$ is a random variable with finite mean $\mu_X$, $\forall a > 0, P(|X - \mu_X| \ge a)\le \frac{Var[X]}{a^2}$

(CHERNOFF's INEQUALITY): $X$ is random variable with MGF $m_X(t)$, $\forall a, P(X \ge a)\le e^{-at}m_X(t)$

(CAUCHY-SCHWARTZ INEQUALITY): $|Cov(X,Y)| \le \sqrt{Var[X]Var[Y]}$

• prove by correlation definition: $|Corr(X, Y)| = |\frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}}| \le 1$

(JENSEN's INEQUALITY): $X$ is random variable with a convex function $f$ s.t. $\mathbb{E}[f(x)]$ is finite, then $f(\mathbb{E}[X]) \le \mathbb{E}[f(X)]$, fliped inequality is true when $f$ is concave.

• $f$ is linear cause equal.