Optimization Models
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1. Introduction
1.1. Overview
1.2. Optimization Models
2. Linear Algebra
2.1. Vectors
2.2. Matrices
2.3. Linear Equations
2.4.  Least-Squares
2.5. Eigenvalues
2.6  Singular Values
3. Convex Models
3.1. Convexity
3.2. LP and QP
3.3. SOCP
3.4. Robust LP
3.5. GP
3.6. SDP
3.7. Non-Convex Models
4. Duality
4.1. Weak Duality
4.2. Strong Duality
4.3.  Applications
5. Case Studies
5.1. Senate Voting
5.2. Antenna Arrays
5.3. Localization
5.4. Circuit Design

Sample and weighted mean, expected value

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

hat{x} := frac{1}{n} ( x_1 + ldots + x_n) .

The sample average can be interpreted as a scalar product:

hat{x} = p^Tx,

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

More generally, for any vector p in mathbf{R}^n, with p_i ge 0 for every i, and p_1+ldots+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,ldots,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 mathbf{E}_p(X), or mathbf{E}(X) if the distribution p is clear from context.