Mean

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

$\mu =\mathbb{E}[X]=⟨x⟩=\bar{x}$

$\mathbb{E}[X]=\sum _{x\in S}x{p}_X(x)$

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

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

• Expected value of a function of a random variable

$\mathbb{E}[X]={\int }_{-\infty }^{\infty }{f}_X(x)\mathrm{d}x$

• Continuous version
• $f_X(x)$ is the PDF

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

• Linearity of Expectation

L2 Norm Proof