2 Introduction to statistics

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2 Recap of Statistics Basic Concepts In this document we review some preliminary concepts of statistics used in what fol- lows of the course and how statistical tools can be used in MATLAB . After that, we briefly mention some of the most known optimization techniques. This is not supposed to be an exhaustive document on the two topics, for more information refer to: • Bishop, Christopher M., “Pattern recognition and machine learning”,2006, Springer; • Montgomery, Douglas C., George Runger C., “Applied statistics and probability for engineers”, 2010, John Wiley & Sons. 2.1 Discrete Random Variables A discrete random variableX is a variable with values in a discrete set E whose value is determined by a stochastic phenomenon, i.e., we are not able to predict its value even if we are given precise information about the phenomenon. For instance, consider a20- faced dice: the event of trowing it can be modeled as a random variableX taking values in the finite set of events E ={1,..., 20}, since we are not able to predict precisely which value might occur, even if we are given all the characteristics of the dice (e.g., dimensions, initial position, speed). To properly model this phenomenons, we define a probability function P : E → [0, 1] which tells you how often the eventi belonginf to a discrete set of eventsE occurs (e.g., probability that by trowing the dice you get 3) as: P(X =i) := |i| |E|, where|i| is the measure of the favorable events set and|E| is the measure of set of the possible events. For instance for the dice we have: P(X =i) = 1 20, since all the faces have the same probability to occur. In this case we have some prop- erties that the probability function should follow: 1 2 Recap of Statistics Basic Concepts • 0≤ P(X = i)≤ 1: an event can occur at least with…