Random Variables and Stochastic Processes 0903720 Dr Ghazi
Random Variables and Stochastic Processes – 0903720 Dr. Ghazi Al Sukkar Email: ghazi. alsukkar@ju. edu. jo Office Hours: will be posted soon Course Website: http: //www 2. ju. edu. jo/sites/academic/ghazi. alsukkar Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul http: //faraday. ee. emu. edu. tr/ee 571/ EE 720 1
Relations of Events q Subset An event E is said to be a subset of the event F if, whenever E occurs, F also occurs. E F * q Equality Events E and F are said to be equal if the occurrence of E implies the occurrence of F, and vice versa. E=F q Intersection * The following three statements are always satisfied: A ⊂ S, ∅ ⊂ A and A ⊂ A EE 720 2
Relations of Events (Cont’d) q Union q Complement q Difference An event is called the difference of two events E and F if it occurs whenever E occurs but F does not, and is denoted by E F. Notes: EC = S E and E F = E FC EE 720 3
Relations of Events (Cont’d) q Certainty An event is called certain if it its occurrence is inevitable. The sample space is a certain event. q Impossibility An event is called impossible if there is certainty in its non-occurence. The empty set is an impossible event. q Mutually Exclusiveness If the joint occurrence of two events E and F is impossible, we say that E and F are mutually exclusive (disjoint). That is, E F = . EE 720 4
Venn Diagrams of Events E F S S E E F F S S F F E EE G (EC G) F EC EE 720 5
Examples q Example 1. 7 At a busy international airport, arriving planes land on a first-come first-served basis. Let E = there at least 5 planes waiting to land, F = there at most 3 planes waiting to land, H = there are exactly 2 planes waiting to land. Then EC is the event that at most 4 planes are waiting to land. FC is the event that at least 4 planes are waiting to land. E is a subset of FC. That is, E FC = E H is a subset of F. That is, F H = H E and F, E and H are mutually exclusive. F HC is the event that the number of planes waiting to land is 0, 1, or 3. EE 720 6
Useful Laws q Commutative Laws: E F = F E, E F = F E q Associative Laws: E (F G) = (E F) G, E (F G) = (E F) G q Distributive Laws: (E F) H = (E H) (F H), (E F) H = (E H) (F H) q De Morgan’s Laws: (E F)C = EC FC, EE 720 7
q De Morgan’s Second Laws: (E F)C = EC FC, EE 720 8
1. 3 Axioms of Probability q Definition: Probability Axioms EE 720 9
Properties of Probability q The probability of the empty set is 0. That is, P( ) = 0 (something has to happen). q Countable additivity & finite additivity q The probability of the occurrence of an event is always some number between 0 and 1. That is, 0 P(A) 1. q Probability is a real-value, nonnegative, countably additive set function. EE 720 10
Field • EE 720 11
Examples Let P be a probability defined on a sample space S. For events A of S define Q(A) = [P(A)]2 and R(A) = P(A)/2. Is Q a probability on S ? Is R a probability on S ? Why or why not? Sol: EE 720 12
Examples q Example 1. 9 A coin is called unbiased or fair if, whenever it is flipped, the probability of obtaining heads equals that of obtaining tails. When a fair coin is flipped, the sample space is S = {T, H}. Since {H} and {T} are equally likely & mutually exclusive, 1 = P(S) = P({T, H}) = P({T}) + P({H}). Hence, P({T}) = P({H}) = 1/2. When a biased coin is flipped, and the outcome of tails is twice as likely as heads. That is, P({T}) = 2 P({H}). Then 1 = P(S) = P({T, H}) = P({T}) + P({H}) =3 P({H}). Hence, P({H}) = 1/3 and P({T}) = 2/3. EE 720 13
Theorem 1. 1 (Classical Definition of Probability) Let S be the sample space of an experiment. If S has N points that are equally likely to occurs, then for any event A of S, where N(A) is the number of points of A. EE 720 14
Examples q Example 1. 10 Let S be the sample space of flipping a fair coin three times and A be the event of at least two heads; then S={ } and A = { }. So N = and N(A) =. The probability of event A is. P(A) = N(A)/N =. EE 720 15
Examples q Example 1. 11 An elevator with 2 passengers stops at the second, third, and fourth floors. If it is equally likely that a passenger gets off at any of the 3 floors, what is the probability that the passengers get off at different floors? Sol:Let a and b denote the two passenger and a 2 b 4 mean that a gets off at the 2 nd floor and b gets off at the 4 th floor. So S = { } and A = { }. So N = and N(A) =. The probability of event A is. P(A) = N(A)/N =. EE 720 16
Examples q Example 1. 12 A number is selected at random from the set of integers {1, 2, …, 1000}. What is the probability that the number is divisible by 3? Sol: Let A be the set of all numbers between 1 and 1000 that are divisible by 3. N = 1000, and N(A) = . Hence, the probability of event A is. P(A) = N(A)/N =. EE 720 17
Basic Theorems S q Theorem 1. 2 For any event A, P(AC) = 1 P(A). B A B-A q Theorem 1. 3 If A B, then P(B A) = P( B AC ) = P(B) P(A). Corollary: If A B, then P(A) P(B). This says that if event A is contained in B then occurrence of B means A has occurred but the converse is not true. EE 720 18
S A Ans: 0. 65 EE 720 A B B 19
Examples Ans: 0. 61 EE 720 20
Inclusion-Exclusion Principle q For 3 events q For n events q Theorem 1. 5 EE 720 21
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