ch1-part 1
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Random Variables
Chapter 1 Part 1
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Sample Space and Events
ExperimentsAny process of trial and observation
Random experiments : an experiment whose outcome is uncertain
Sample space The collection of possible elementary outcomes
Sample points : the elementary outcomes of an experiment denoted by wi, i=1,2,
Event any one of a number of possible outcomes of an experiment
a subset of the sample space
Example if we toss a die, the sample space is
the outcome of the toss of a die is an even number
three coin-tossing experiment
the event one head and two tails
nwwwS ,,, 21 Sample space Sample points
6,5,4,3,2,1S 6,4,2E
TTTTTHTHTTHHHTTHTHHHTHHHS ,,,,,,, TTHTHTHTTE ,,
Elementary outcomes
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Sample Space and Events
In a single coin toss experiment, sample space
the event that a head appears on the toss
the event that a tail appears on the toss
If we toss a coin twice sample space
the event that a head appears on the 2nd toss
If we toss a die twicesample space
the event that the sum of the two tosses is 8
If we measure the lifetime of an electronic component Sample space
the event that the lifetime is not more that 7 hours
THS ,
HE
TE
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Sample Space and Events
1.2 Sample Space and Events algebra of events
union of events A and B intersection of events A and B
Union of events A and B : the event that consists of all sample points that are either in A or in B or in both A and B
Intersection of events A and B : the event that consists of all sample points that are in both A and B
Mutually exclusive : if their intersection contains no sample point.
Difference of events A and B : the event that all sample points are in A but not in B.
BAC BAD
BAC
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Definition of Probability
Three different kinds of definitions for Probabilityaxiomatic, relative-frequency, classical definitions
1.3.1 Axiomatic DefinitionFor 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
1)(0 AP
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nm
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Axiom 1
Axiom 2
Axiom 3
; work with nonnegative numbers
; sample space itself is an event,
it should have the highest probability
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.
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Definition of probability ExampleObtaining a number x by spinning the pointer on a fair wheel of chance that is labeled from 1 to 100 points.
Sample space
The probability of the pointer falling between any two numbers
Consider events
1)( SP
Axiom 1
Axiom 2
Axiom 3
}1000|{ xxS12 xx
100/)()( 1221 xxxxxP
}{ 21 xxxA
0and100 12 xx
},{anyfor 1 nnn xxxA
NAP n /1)(
11
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1)(
11
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n
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n
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n
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NAP
SPAP
Nnxn /100)(
; for all x1, x2
Break the wheels periphery into N continuous segments, n=1,2,N with x0=0
1)(0 AP
unbiased
Definition of Probability
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1.3.2 Relative frequency definition
Probability as a relative frequency
Flip a coin : heads show up nA times out of the n flips
Probability of the event heads
Statistical regularity : relative frequencies approach a fixed value (a probability) as nbecomes large.
1.3.3 Classical Definition
Probability as a classical definition
This probability is determined a priori without actually performing the experiment.
n
nAP A
nlim)(
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Definition of Probability
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First Die
Se
con
d D
ie
1
2
3
4
5
6
1 2 3 4 5 6
(1,1)
(1,2)
(1,4)
(1,3)
(1,5)
(1,6)
(2,1) (3,1) (4,1) (5,1)(6,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
A1
A2
C
Example Tossing two dice
Figure 1.1 Sample Space for Example 1.1
- Sample space : 62=36 points
- For each possible outcome,
a sum having values from 2 to 12
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},12{
}11{}7{
}11{},7{ 21
evendicebothD
diedieC
sumorsumB
sumAsumA
stnd
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Definition of Probability
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Summary - Mathematical model of Experiments
A real experiment is defined mathematically by three thing
1. Assignment of a sample space
2. Definition of events of interest
3. Making probability assignment to the events such that the axioms are satisfied
Generally, it is not easy to construct correct mathematical model
A die worn out
Definition of Probability
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Exercise conditional probability
Example
80 resistors in a box : 10W -18, 22W -12, 27W -33, 47W -17, draw out one resistor, equally likely
Suppose a 22W is drawn and not replaced. What are not the probabilities of drawing a resistor of any one of four values?
80/17)47(80/33)27(
80/12)22(80/18)10(
drawPdrawP
drawPdrawP
79/17)22|47(
79/33)22|27(
79/11)22|22(
79/18)22|10(
drawP
drawP
drawP
drawP
The concept of conditional probability is needed.
Definition of Probability
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# Homework for reading : 1.4 Application of Probability
1.4.1 Reliability Engineering
1.4.2 Quality Control (QC)
1.4.3 Channel Noise
1.4.4 System simulation
random # of generation that can be used to represent events such as arrival of customers at a bank in the sytem being modeled
Our main work will be focused on the area
Applications of Probability
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Definitions
Set : a collection of objects A (capital letter)
Objects : Elements of the set a (small letter)
If a is an element of set A :
If a is not an element of set A :
Methods for specifying a set
Tabular method
Ex) {6, 7, 8, 9}
Rule method
Ex) {integers between 5 and 10}, {i | 5 < i < 10, i an integer}
Set
Countable, uncountable
Finite, infinite
Null set(=empty) :
a subset of all other sets
countably infinite set
Aa
Aa
Ex) - a set of voltages- a set of airplanes- a set of chairs- a set of sets
Elementary Set Theory
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Definitions
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
BA
BA
Set Definitions
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Example
A : Tabularly specified, countable, and finite
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
}5.85.0{
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}7,5,3,1{
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}14,12,10,8,6,4,2{
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D
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Set Definitions
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Universal set
The largest set or all-encompassing set of objects under discussion in a given situation
Power set
Power set of A : the set of all subsets of a set A, s(A)
Example
A = {a,b} s(A) = {{a}, {b}, {a,b}, }
Cardinality
Cardinality of A : the number of members of a set A, |A|.
Example
A = {a,b} |A| = 2
|s(A)| = 2n when |A| = n
Set Definitions
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Rolling a die (Example 1.1-2)
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 subset 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}
Example : Tossing a token twice
S = {TT, HT, TH, HH} 24=16 number of subsets exist
Example : Rolling a die
S = {1, 2, 3, 4, 5, 6} 26=64 number of subsets exist
Set Definitions
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Problems
Specify the following sets by the rule method.
A={1,2,3} -> A={k | 0 < k < 4}
B={8,10,12,14} -> B={k | 6 < k C={2k-1 | k is the positive integer}
State every possible subset of the set of letters {a,b,c,d}
{}, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}. {b,c,d}, {a,b,c,d} -> Total 16 number of subsets
A random noise voltage at a given time may have any value from -10 to 10V.
(a) What is the universal set describing noise voltage?
-> S={s | -10s10}
(b) Find a set to describe the voltages available from a half-rectifier for positive voltages that has a linear output-input characteristic.
-> V={s | 0s10}
(c) Repeat parts (a) and (b) if a DC voltage of -3V is added to the random noise.
-> S={ s | -13s7}, V={s |0s7}
Set Definitions Problem Solving
R
R
R R
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Venn Diagram
Equality : A=B
Two sets are equal if all elements in A are present in B and all elements in B are in A
That is, if
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
ABandBA BA
C is disjoint from both A and B
B is a subset of A
}5.20.1{},6.16.0{ bBaA
}5.26.1{},0.16.0{ bABaBA ABBA
Universal set
Set Operations
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Union and intersection
Union (Sum) :
The union (call it C) of two sets A and B
The set of all elements of A or B or both
Intersection (Product) :
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 (M.E.) sets A and B,
The union and intersection of N sets An, n=1,2,,N
Complement
The complement of the set A is the set of all elements not in A
BAC
BAD
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ASA
AAandSAASS ,,,
Set Operations
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Example
Union (Sum) and Intersection (Product)
Complement
}8,7,6,4,3,1{
}11,10,9,8,7,6,2{
}12,5,3,1{
}12integers1{
C
B
A
S
}11,10,9,8,7,6,4,3,2,1{
}12,8,7,6,5,4,3,1{
}12,11,10,9,8,7,6,5,3,2,1{
CB
CA
BA
}8,7,6{
}3,1{
CB
CA
BA
}12,11,10,9,5,2{
}12,5,4,3,1{
}11,10,9,8,7,6,4,2{
C
B
A
Set Operations
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Duality Principle
If in an identity we replace unions by intersections, intersections by unions,
Example
)()()(
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CABACBA
CABACBA
}43{
}10,8,6,2{
}6,4,2,1{
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B
A
}6,4,2{)()(
}6,4,2{)(
CABA
CBA
}4{
}6,2{
}10,8,6,43,2{
CA
BA
cCB
)()()( CABACBA
Set Operations
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Algebra of Sets
Commutative law
Distributive law
Associative law
ABBA
ABBA
)()()(
)()()(
CABACBA
CABACBA
CBACBACBA
CBACBACBA
)()(
)()(
Set Operations
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De Morgans law
The complement of a union (intersection) of two sets A and B equals the intersection (union) of the complements and
Example
A B
BABA
BABA
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}225{},162{
}242{
bBaA
sS
}2416,52{
}2422,52{
},2416{
ccBAC
aaBSB
aASA
BABA )(
}2416,52{
}165{
ccBAC
cBABAC
Set Operations
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Problems
Show that C A if C B and B A.
Explain it by using Ven diagram
Two sets are given by A={-6, -4, -0.5, 0, 1.6, 8} and B={-0.5,0,1,2,4}. Find:
(a) A-B -> {-6, -4, 1.6, 8}
(b) B-A -> {1, 2, 4}
(c) AB -> {-6, -4, -0.5, 0, 1, 1.6, 2, 4, 8}
(d) AB-> {-0.5, 0}
1.2-4. Using Venn diagrams for three sets A,B,C, shade the areas corresponding to the sets:
(a) (AB)-C (b) A-B (c) C-(AB)
Set Operations Problem Solving
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Problems
Sketch a Venn diagram for three events where AB0, BC0, CA0, but ABC=0.
Show the equations of Venn diagrams
Sets A={1s14}, B={3,6,14}, and C={2
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Properties of Probability
Properties of Probability
1. The probability of the complement of A is one minus the probability of A.
2. The null event has probability zero.
3. If A is a subset of B, the probability of A is at most the probability of B.
4. P(A)1 the probability of event A is at most 1.
5.
6.
7.
8. Generalization of Property 7
)(1)( APAP
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)()( BPAPBA
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)()()()()( BAPBAPAPBABAA
)()()()( BAPBPAPBAP Joint Probability )( BAP
)()()()()()( BPAPBAPBPAPBAP
The probability of the union of two events never exceeds the sum of the event probabilities.