¡Descarga Probability Theory: Understanding Random Experiments and Events y más Apuntes en PDF de Administración de Empresas solo en Docsity! Statistics I Topic 2 - Probability Theory. Probability Theory is a set of tools and techniques to deal with random phe- nomena Any experiment of phenomenon that satisfies the three properties below is a random experiment 1. The possible outcomes of the experiment are known before hand. 2. It is impossible to foresee the final outcome of the experiment before con- ducting it. 3. Different trials of the experiment, conducted under the same conditions, pro- duce different outcomes. Year 2011-2012 1 Statistics I Probability Theory Exemple 1 If we toss a (fair) coin, 1. We know that there are two possible outcomes only: Heads or Tails 2. We cannot foresee what the actual outcome will be 3. Different tosses of the coin will produce different outcomes Year 2011-2012 2 Statistics I Probability Theory Exemple 3 A coin is tossed repeatedly until “Heads” occurs for the first time Ω = {c,+c,++ c,+++c,++++ c,+++++c, . . .} This is an infinite countable sample space Exemple 4 Randomly select a real number between 0 and 1 Ω = {x ∈ℜ/x ∈ [0,1]} This is an infinite uncountable sample space Year 2011-2012 5 Statistics I Probability Theory • In many cases we might find that, apparently, there are different ways of specifying the sample space associated to an experiment. If, for instance, we toss two dice and are interested in the sum of the numbers that result, we might thing that the sample space is: Ω = {2,3,4,5,6,7,8,9,10,11,12} for this are the possible results of the sum of the outcomes of two dice. • Nevertheless, this sample space is not quite right because it “hides” some information regarding the outcome of the experiment. • In particular, it does not tell what is the outcome in each dice. Although this does not seem to be relevant now, it will turn out to be extremely important when we want to find the probability of the different outcomes. Year 2011-2012 6 Statistics I Probability Theory • In this sense, a better specification of the sample space that corresponds to this experiment is: Ω = {(1,1), (1,2), . . . ,(1,6), (2,1), (2,2), . . . ,(6,5),(6,6)} • As a general rule, for a given random experiment, the sample space must provide as much information as possible about the outcomes of such exper- iment. Year 2011-2012 7 Statistics I Probability Theory Exemple 9 A dice is tossed. The four subsets of Ω below correspond to the events: “The outcome is an even number”, “The outcome is less than 3”, “The outcome is not 6” and “The outcome is 1” respectively (notice that the event “The outcome is 1” is an elementary event): A = {2,4,6} B = {1,2} C = {1,2,3,4,5} D = {1} Year 2011-2012 10 Statistics I Probability Theory 2.1.2 σ -algebra of events (A ). Definició 10 Given a random experiment and its associated sample space Ω, the set of all the possible events (that is, the set of all the subsets of Ω is called the σ -algebra of events and is denoted with the letter A . Exemple 11 A dice is tossed Ω = {1,2,3,4,5,6} A = {Ω, ,{2,4,6},{1,2},{1,2,3,4,5},{1}, . . . ,etc, . . .} That is, A = {A/A⊆Ω} Year 2011-2012 11 Statistics I Probability Theory In the language of Set Theory, A is the Power Set of Ω, P(Ω). Definició 12 We say that an event occurs (or has occurred) if the outcome of the experiment says so. That is, if the outcome of the experiment is ω j, the event A occurs (or has occurred) if ω j ∈ A. Exemple 13 In example 9, if the outcome of the experiment is 4, we have A occurs since 4 ∈ A. B does not occur since 4 /∈ B. C occurs since 4 ∈C. D does not occur since 4 /∈ D. Year 2011-2012 12 Statistics I Probability Theory 2.1.4 Operations with events (i) Complement.- Ā is the event that occurs if, and only if, A does not occur. (ii) Union.- A∪B is the event that occurs if, and only if, A orB (or both) occur. (iii) Intersection.- A∩B is the event that occurs if, and only if, A and B occur simultaneously. (iv) Difference.- A\B is the event that occurs if, and only if, A occurs and B does not occur. Year 2011-2012 15 Statistics I Probability Theory 2.1.5 Venn diagrams To deal an operate with events the so-called Venn diagrams are very useful. A Venn diagram, which is a common tool in set theory, consists of a rectan- gle that represents the sample space Ω and a collection of figures inside that represent the different events of interest (see figure 1) Figure 1: Venn diagram Year 2011-2012 16 Statistics I Probability Theory Exemple 14 A dice is tossed. The events A =“The outcome is an even num- ber”, B =“The outcome is less than 3”, C =“The outcome is not 6” and D =“The outcome is 1” are represented by means of Venn diagrams in figure 2 Figure 2: Venn diagram for example 14 Year 2011-2012 17 Statistics I Probability Theory (ii) Intersection: Given two events A and B, the intersection event A∩B is the area where the two events overlap (shadowed area in figure 5) Figure 5: Intersection Year 2011-2012 20 Statistics I Probability Theory (ii) Difference: Given two events A and B, the difference event A\B is the part of the area of A that is not in B (shadowed area in figure 6) Figure 6: Difference Year 2011-2012 21 Statistics I Probability Theory 2.1.6 Properties of events. The properties below, which can be verifies using Venn diagram, are very useful for doing operations with events (i) Distributive laws: A∩ (B∪C) = (A∩B)∪ (A∩C) A∪ (B∩C) = (A∪B)∩ (A∪C) Year 2011-2012 22
