HANDOUT 1 PROBABILITY AND DISTRIBUTION THEORY I

Definition 1 A random experiment is an experiment satisfying the fol-lowing three conditions: (i) All possible distinct outcomes are known a priori; (ii) In any particular trial the outcome is not known a priori; (iii) It can be repeated under identical conditions. Definition 2 The sample space Ω is defined to be the set of all possible outcomes of the random experiment. Example 3 When throwing a dice, the sample space is Ω = {1 2 3 4 5 6} Example 4 Consider the sum of points when throwing two dices, the sample space will be Ω = {2 3 4 5 6 7 8 9 10 11 12} Definition 5 An elementary event is the element of the sample space Ω. Example 6 When throwing a dice, the element {1} is an elementary event. Definition 7 An event is a subset of the sample space Ω. Every subset is an event. Thus an event may be an empty set, a proper subset of the sample space, or the sample space itself. An elementary event is an event while an event may not be an elementary event. 1 Example 8 Consider the sum of points when throwing two dices, the event that the sum is an even number will be = {2 4 6 8 10 12} Example 9 The event that the sum is bigger than 13 will be an empty set , we call it a null event. Example 10 The event that the sum is smaller than 13 will be {2 3 4 5 6 7 8 9 10 11 12}, or equal the sample space. Definition 11 The collection = of subsets of Ω is called −algebra if it satisfies the following properties: () Ω ∈ =, () ∈ = ⇒ ∈ = (closure under complementation) where refers to the complement of with respect to Ω. () ∈ =, = 1 2 ⇒ ∪∞=1 ∈ = (closure under countable union