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확률및공학통계 (Probability and Engineering Statistics)

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, 4 th Ed. 저자 : A. Papoulis, S. U. Pillai.

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확률및공학통계 (Probability and Engineering Statistics)

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  1. 확률및공학통계(Probability and Engineering Statistics) 이시웅

  2. 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

  3. 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)

  4. 1.1 Set Definition • Set : a collection of objects - A • Objects: Elements of the set - a • If a is an element of set A : • If a is not an element of set A : • Methods for specifying a set • Tabular method • Rule method • Set • Countable, uncountable • Finite, infinite • Null set(=empty) : Ø : a subset of all other sets • Countably infinite

  5. 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

  6. 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 • 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

  7. 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}

  8. S B C A 1.2 Set Operations • Venn Diagram • 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.

  9. 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 = {1.0b2.5} • A-B = {0.6 < c < 1.0} • B-A = {1.6 < d  2.5}

  10. Union (Sum): C = AB • 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 = AB • 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, AB = Ø • The union and intersection of N sets An, n = 1,2,…,N :

  11. Complement : • The complement of the set A is the set of all elements not in A

  12. S 5,12 A 1,3 C 4 6,7,8 2,9,10,11 B • Example • Applicable unions and intersections • Complements

  13. Algebra of Sets • Commutative law: • Distributive law • Associative law

  14. De Morgan’s Law • The complement of a union (intersection) of two sets A and B equals the intersection (union) of the complements and

  15. Example 1.2-2

  16. Example 1.2-3

  17. 1.3 Probability Introduced Through Sets and Relative Frequency • Definition of probability • Set theory and fundamental axioms • Relative frequency • Experiment : Rolling a single die • Six numbers : 1/6 • Sample space (S) • The set of all possible outcomes in any experiments Universal set • Discrete, continuous • Finite, infinite All possible outcomes likelihood

  18. Mathematical model of experiments • Sample space • Events • 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 • events • Discrete or continuous

  19. 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: , event: A= {7.4<a<7.6} • Continuous sample space and continuous event • Sample space: , event A = {6.1392} • Continuous sample space and discrete event

  20. 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 • S:certain event, Ø: impossible event 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

  21. Obtaining a number x by spinning the pointer on a “fair” wheel of chance that is labeled from 0 to 100 points • Sample space • The probability of the pointer falling between any two numbers : • Consider events • Axiom 1: • Axiom 2: • Axiom 3: Break the wheel’s periphery into N continuous segments, n=1,2,…,N with x0=0 , for any N,

  22. If the interval is allowed to approach to zero (->0), the probability • Since in this situation, • 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

  23. Mathematical Model of Experiments • A real experiment is defined mathematically by three things • Assignment of a sample space • Definition of events of interest • Making probability assignments to the events such that the axioms are satisfied

  24. 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

  25. Assigning probabilities to these events • Define 36 elementary event, i = row, j = column • Aij:Mutually exclusive events-> axiom 3 • The events A, B, and C are simply the unions of appropriate events

  26. Probability as a Relative Frequency • Flip a coin: heads shows up nH times out of the n flips • Probability of the event “heads”: • Relative frequency: • Statistical regularity: relative frequencies approach a fixed value(a probability) as n becomes large

  27. Example 1.3-3 • 80 resistors in a box:10-18, 22-12, 27-33, 47-17, draw out one resistor, equally likely • Suppose a 22 is drawn and not replaced. What are now the probabilities of drawing a resistor of any one of four values?

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