Mean

#Math
Central tendency measure that minimizes the sum of L2 norms between it and other data points

$\displaystyle \mu=\mathbb{E}[X]={\left\langle{x}\right\rangle}=\bar{x}$

$\displaystyle \mathbb{E}[X]=\sum_{x \in S}xp_{X}(x)$

• Discrete version
• $\displaystyle \mathbb{E}[X]$ is the expected value of random variable $\displaystyle X$
• $\displaystyle p_{X}(x)$ is the PMF of random variable $\displaystyle X$ for a value $\displaystyle x$
• $\displaystyle S$ is the set of values of our random variable $\displaystyle X$

$\displaystyle \mathbb{E}[g(X)]=\displaystyle\sum_{x\in S}g(x)p_X(x),~g:S\mapsto \mathbb{R}$

• Expected value of a function of a random variable

$\displaystyle \mathbb{E}[X]=\int_{-\infty}^{\infty} f_{X}(x) , \mathrm{d}x$

• Continuous version
• $\displaystyle f_{X}(x)$ is the PDF

$\displaystyle \mathbb{E}[ag(X)+bh(X)]=a\mathbb{E}[g(X)]+b\mathbb{E}[h(X)]$

• Linearity of Expectation

L2 Norm Proof