Random Variables and Stochastic Processes – Dr. Ghazi Al Sukkar Office Hours: will be posted soon Course Website: EE 7201 Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul /

Relations of Events EE 7202  Subset An event E is said to be a subset of the event F if, whenever E occurs, F also occurs.  E  F *  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  Intersection * The following three statements are always satisfied: A ⊂ S, ∅ ⊂ A and A ⊂ A

Relations of Events (Cont’d) EE 7203  Union  Complement  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: E C = S  E and E  F = E  F C

Relations of Events (Cont’d) EE 7204  Certainty An event is called certain if it its occurrence is inevitable. The sample space is a certain event.  Impossibility An event is called impossible if there is certainty in its non-occurence. The empty set is an impossible event.  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 = .

Venn Diagrams of Events EE 7205 E F S EFEF E F E S ECEC E F S (ECG)  F(ECG)  F E F GE F S EFEF

Examples EE 7206  Example 1.7 At a busy international airport, arriving planes land on a first-come first-served basis. Let E = there are at least 5 planes waiting to land, F = there are at most 3 planes waiting to land, H = there are exactly 2 planes waiting to land. Then E C is the event that at most 4 planes are waiting to land. F C is the event that at least 4 planes are waiting to land. E is a subset of F C. That is, E  F C = E H is a subset of F. That is, F  H = H E and F, E and H are mutually exclusive. F  H C is the event that the number of planes waiting to land is 0, 1, or 3.

Useful Laws EE 7207  Commutative Laws: E  F = F  E, E  F = F  E  Associative Laws: E  (F  G) = (E  F)  G, E  (F  G) = (E  F)  G  Distributive Laws: (E  F)  H = (E  H)  (F  H), (E  F)  H = (E  H)  (F  H)  De Morgan’s Laws: (E  F) C = E C  F C,

 De Morgan’s Second Laws: EE 7208 (E  F) C = E C  F C,

1.3 Axioms of Probability EE 7209  Definition ： Probability Axioms

Properties of Probability EE  The probability of the empty set  is 0. That is, P(  ) = 0 ( something has to happen ).  Countable additivity & finite additivity  The probability of the occurrence of an event is always some number between 0 and 1. That is, 0  P(A)  1.  Probability is a real-value, nonnegative, countably additive set function.

Field EE 72011

Examples EE 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 ：

Examples EE  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}) = 2P({H}). Then 1 = P(S) = P({T,H}) = P({T}) + P({H}) =3P({H}). Hence, P({H}) = 1/3 and P({T}) = 2/3.

Theorem 1.1 EE (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.

Examples EE  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 =.

Examples EE  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 =.

Examples EE  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 =.

Basic Theorems EE  Theorem 1.2 For any event A, P(A C ) = 1  P(A).  Theorem 1.3 If A  B, then P(B  A) = P( B  A C ) = P(B)  P(A). S A B-A B 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 Ans ： 0.65 S AB ABAB

Examples EE Ans ： 0.61

Inclusion-Exclusion Principle EE  For 3 events  For n events  Theorem 1.5

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