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확률및공학통계 (Probability and Engineering Statistics) 이시웅 교재 • 주교재 – 서명 : Probability, Random Variables and Random Signal Principles – 저자 : P. Z. Peebles, 역자 : 강훈외 공역 • 보조교재 – 서명 : Probability, Random Variables and Stochastic Processes, 4th Ed. – 저자 : A. Papoulis, S. U. Pillai Introduction to Book • Goal – Introduction to the principles of random signals – Tools for dealing with systems involving such signals • Random Signal – A time waveform that can be characterized only in some probabilistic manner – Desired or undesired waveform(noise) 1.1 Set Definition • • • • • Set : a collection of objects - A Objects: Elements of the set - a If a is an element of set A : a A If a is not an element of set A : a A Methods for specifying a set 1. Tabular method 2. Rule method • Set – – – – Countable, uncountable Finite, infinite Null set(=empty) : Ø : a subset of all other sets Countably infinite • A is a subset of B : : If every element of a set A is also an element in another set B, A is said to be contained in B • A is a proper subset of B : : If at least one element exists in B which is not in A, • Two sets, A and B, are called disjoint or mutually exclusive if they have no common elements • • • • • • • • • • A {1,3,5,7} B {1,2,3, } D {0.0} E {2,4,6,8,10,12,14} C {0.5 c 8.5} F {5.0 f 12.0} A : Tabularly specified, countable B : Tabularly specified, countable, and infinite C : Rule-specified, uncountable, and infinite D and E : Countably finite F : Uncountably infinite D is the null set? A is contained in B, C, and F C F , D F and E B B and F are not subsets of any of the other sets or of each other A, D, and E are mutually exclusive of each other • Universal set : S – The largest set or all -encompassing set of objects under discussion in a given situation • Example 1.1-2 – Rolling a die • S = {1,2,3,4,5,6} • A person wins if the number comes up odd : A ={1,3,5} • Another person wins if the number shows four or less : B = {1,2,3,4} • Both A and B are subsets of S – For any universal set with N elements, there are 2N possible subsets of S • Example : Token – S = {T, H} – {}, {T}, {H}, {T,H} 1.2 Set Operations • Venn Diagram S B A C – C is disjoint from both A and B – B is a subset of A • Equality : A = B – Two sets are equal if all elements in A are present in B and all elements in B are present in A; that is, if A B and B A. • Difference : A - B – The difference of two sets A and B is the set containing all elements of A that are not present in B – Example: A = {0.6< a 1.6}, B = {1.0b2.5} • A-B = {0.6 < c < 1.0} • B-A = {1.6 < d 2.5} • Union (Sum): C = AB – The union (call it C) of two sets A and B – The set of all elements of A or B or both • Intersection (Product) : D = AB – The intersection (call it D) of two sets A or B – The set of all elements common to both A and B – For mutually exclusive sets A and B, AB = Ø • The union and intersection of N sets An, n = 1,2,…,N : C A1 A2 D A1 A2 AN AN N An , n 1 N An n 1 • Complement : – The complement of the set A is the set of all elements not in A – AS A – S , S , A A S , and A A • Example S {1 integers 12} A {1,3,5,12} B {2,6,7,8,9,10,11} C {1,3,4,6,7,8} – Applicable unions and intersections A B {1,2,3,5,6,7,8,9,10,11,12} A B A C {1,3} A C {1,3,4,5,6,7,8,12} B C {6,7,8} B C {1,2,3,4,6,7,8,9,10,11} – Complements A {2,4,6,7,8,9,10,11} B {1,3,4,5,12} C {2,5,9,10,11,12} S 5,12 1,3 A C 4 6,7,8 2,9,10,11 B • Algebra of Sets – Commutative law: – Distributive law – Associative law A B B A A B B A A ( B C ) ( A B) ( A C ) A ( B C ) ( A B) ( A C ) ( A B) C ( A B) C A B C ( A B) C ( A B) C A B C • De Morgan’s Law – The complement of a union (intersection) of two sets A and B A equals the intersection (union) of the complements and B ( A B) A B ( A B) A B • Example 1.2-2 S {2 s 24} A {2 a 16}, B {5 b 22} C A B A B {5 c 16} C A B {2 c 5, 16 c 24} A S A {16 a 24}, B S B {2 a 5, 22 a 24} C A B {2 c 5, 16 c 24} ( A B) A B • Example 1.2-3 A {1,2,4,6} B {2,6,8,10} B C {2, 3 c 4, 6,8,10} A B {2,6} C {3 c 4} A C {4} A ( B C ) {2,4,6} ( A B) ( A C ) {2,4,6} A ( B C ) ( A B) ( A C ) 1.3 Probability Introduced Through Sets and Relative Frequency • Definition of probability 1. Set theory and fundamental axioms 2. Relative frequency • Experiment : Rolling a single die – Six numbers : 1/6 All possible outcomes likelihood • Sample space (S) – The set of all possible outcomes in any experiments Universal set – Discrete, continuous – Finite, infinite • Mathematical model of experiments 1. Sample space 2. Events 3. Probability • Events – – – – Example : Draw a card from a deck of 52 cards -> “draw a spade” Definition : A subset of the sample space Mutually exclusive : two events have no common outcomes Card experiment • Spades : 13 of the 52 possible outputs • 2 N 252 4.5(1015 ) events – Discrete or continuous – Events defined on a countably infinite sample space do not have to be countably infinite • Sample space: {1, 3, 5, 7, …} event: {1,3,5,7} – Sample space: S {6 s 13} , event: A= {7.4<a<7.6} • Continuous sample space and continuous event – Sample space: S {6 s 13} , event A = {6.1392} • Continuous sample space and discrete event • Probability Definition and Axioms – Probability • To each event defined on a sample space S, we shall assign a nonnegative number • Probability is a function • It is a function of the events defined • P(A): the probability of event A • The assigned probabilities are chosen so as to satisfy three axioms 1. P( A) 0 2. P( S ) 1 S:certain event, Ø: impossible event 3. N N PU An P( An ) if Am An n 1 n 1 for all m n = 1, 2, …, N with N possibly infinite The probability of the event equal to the union of any number of mutually exclusive events is equal to the sum of the individual event probabilities • Obtaining a number x by spinning the pointer on a “fair” wheel of chance that is labeled from 0 to 100 points – Sample space S {0 x 100} – The probability of the pointer falling between any two numbers x2 x1 : ( x2 x1 ) / 100 – Consider events A {x1 x x2} • Axiom 1: x2 100 and x1 0 • Axiom 2: • Axiom 3: Break the wheel’s periphery into N continuous segments, n=1,2,…,N with x0=0 P( An ) 1 / N , for any N, An {xn1 x xn }, xn (n)100 / N N 1 N N PU An P( An ) 1 P( S ) n 1 n 1 n 1 N – If the interval xn xn1 is allowed to approach to zero (->0), the probability P( An ) P( xn ) • Since N in this situation, P( An ) 0 • Thus, the probability of a discrete event defined on a continuous sample space is 0 • Events can occur even if their probability is 0 • Not the same as the impossible event • Mathematical Model of Experiments – A real experiment is defined mathematically by three things 1.Assignment of a sample space 2.Definition of events of interest 3.Making probability assignments to the events such that the axioms are satisfied • Observing the sum of the numbers showing up when two dice are thrown – Sample space : 62=36 points – Each possible outcome: a sum having values from 2 to 12 – Interested in three events defined by A {sum 7}, B {8 sum 11}, C {10 sum} – Assigning probabilities to these events • Define 36 elementary event, i = row, j = column Aij {sum for outcome (i, j ) i j} • P( Aij ) 1 / 36 • Aij: Mutually exclusive events-> axiom 3 • The events A, B, and C are simply the unions of appropriate events 6 6 1 1 P( A) P Ai ,7 i P( Ai , 7 i ) 6 36 6 i 1 i 1 1 1 1 1 P( B) 9 , P(C ) 3 36 4 36 12 1 1 1 5 P( B C ) 2 , P( B U C ) 10 36 18 36 18 • Probability as a Relative Frequency – Flip a coin: heads shows up nH times out of the n flips – Probability of the event “heads”: n lim H P( H ) n n – Relative frequency: nH n – Statistical regularity: relative frequencies approach a fixed value(a probability) as n becomes large • Example 1.3-3 – 80 resistors in a box:10-18, 22-12, 27-33, 47-17, draw out one resistor, equally likely P(draw 10) 18 / 80 P(draw 22) 12 / 80 P(draw 27) 33 / 80 P(draw 47) 17 / 80 – Suppose a 22 is drawn and not replaced. What are now the probabilities of drawing a resistor of any one of four values? P(draw 10 | 22) 18 / 79 P(draw 22 | 22) 11 / 79 P(draw 27 | 22) 33 / 79 P(draw 47 | 22) 17 / 79