Sample and weighted average

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Sample and weighted average

The sample mean (or, average) of given numbers $x_1, \cdots, x_n$ , is defined as

$\begin{align*} \hat{x} &:= \frac{1}{n} (x_1 + \cdots + x_n). \end{align*}$

The sample average can be interpreted as a scalar product:

$\begin{align*} \hat{x} &= p^T x. \end{align*}$

where $x = (x_1, \cdots, x_n)$ is the vector containing the samples, and $p = (1/n){\bf 1}$ , with ${\bf 1}$ the vector of ones.

More generally, for any vector $p \in \mathbb{R}^n$ , with $p_i \ge 0$ for every $i$ , and $p_1 + \cdots + p_n = 1$ , we can define the corresponding weighted average as $p^Tx$ . The interpretation of $p$ is in terms of a discrete probability distribution of a random variable $X$ , which takes the value $x_i$ with probability $p_i$ , $i = 1, \cdots, n$ . The weighted average is then simply the expected value (or, mean) of $X$ under the probability distribution $p$ . The expected value is often denoted ${\bf E}_p(X)$ , or ${\bf E}(X)$ if the distribution $p$ is clear from context.

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