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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 = {1.0b2.5}
• A-B = {0.6 < c < 1.0}
• B-A = {1.6 < d  2.5}
•
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 :
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
– AS 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
PU 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  {xn1  x  xn },
xn  (n)100 / N
N
1
N  N
PU An    P( An )    1  P( S )
 n 1  n 1
n 1 N
– If the interval xn  xn1 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