Download Chap. 3: Probability

© 2003 Pearson Prentice Hall
Probability
3-1
© 2003 Pearson Prentice Hall
Experiments,
Outcomes, & Events
3-2
Experiments & Outcomes
© 2003 Pearson Prentice Hall
1. Experiment

Process of Obtaining an Observation,
Outcome or Simple Event
2. Sample Space (S)

3-3
Collection of All Possible Outcomes
Outcome Examples
© 2003 Pearson Prentice Hall
Experiment
Sample Space
Toss a Coin, Note Face
Head, Tail
Toss 2 Coins, Note Faces HH, HT, TH, TT
Play a Football Game
Win, Lose, Tie
Inspect a Part, Note Quality Defective, OK
Observe Gender
Male, Female
3-4
Events
© 2003 Pearson Prentice Hall
Any Collection of Sample Points
(outcomes)
Simple Event

Collection of outcomes that’s simple to
describe
Compound Event

Collection of outcomes that is described
as unions or intersections of other events
3-5
Event Examples
© 2003 Pearson Prentice Hall
Experiment: Toss 2 Coins. Note Faces.
Event
Sample Space
1 Head & 1 Tail
Heads on 1st Coin
At Least 1 Head
Heads on Both
3-6
Outcomes in Event
HH, HT, TH, TT
HT, TH
HH, HT
HH, HT, TH
HH
© 2003 Pearson Prentice Hall
Sample Space
3-7
© 2003 Pearson Prentice Hall
1.
Visualizing
Sample Space
Listing

S = {Head, Tail}
2.
Venn Diagram
3.
Contingency Table
4.
Decision Tree Diagram
3-8
Venn Diagram
© 2003 Pearson Prentice Hall
Experiment: Toss 2 Coins. Note Faces.
Tail
TH
Outcome
HH
HT
TT
S
S = {HH, HT, TH, TT}
3-9
Sample Space
Event
Contingency Table
© 2003 Pearson Prentice Hall
Experiment: Toss 2 Coins. Note Faces.
2
st
Tail
Total
Head
HH
HT
HH, HT
Tail
TH
TT
TH, TT
Total
HH, TH HT, TT
S = {HH, HT, TH, TT}
3 - 10
Coin
Head
1 Coin
Simple
Event
(Head on
1st Coin)
nd
S
Sample Space
Outcome
Tree Diagram
© 2003 Pearson Prentice Hall
Experiment: Toss 2 Coins. Note Faces.
H
HH
T
HT
H
Outcome
H
TH
T
TT
T
S = {HH, HT, TH, TT}
3 - 11
Sample Space
© 2003 Pearson Prentice Hall
Probabilities
3 - 12
What is Probability?
© 2003 Pearson Prentice Hall
1. Numerical Measure
of Likelihood that
Event Will Occur



P(Event)
P(A)
Prob(A)
1
Certain
.5
2. Lies Between 0 & 1
3. Sum of outcome
probabilities is 1
3 - 13
0
Impossible
Probability
© 2003 Pearson Prentice Hall
P(A)=lim[n(A)/N0]
N0
∞
3 - 14
Many Repetitions!
© 2003 Pearson Prentice Hall
Total Heads /
Number of Tosses
1.00
0.75
0.50
0.25
0.00
0
25
50
75
Number of Tosses
3 - 15
100
125
© 2003 Pearson Prentice Hall
Conditional Probability
3 - 16
Conditional Probability
© 2003 Pearson Prentice Hall
1. Event Probability Given that Another
Event Occurred
2. Revise Original Sample Space to
Account for New Information

Eliminates Certain Outcomes
3. P(A | B) = P(A and B)
P(B)
3 - 17
© 2003 Pearson Prentice Hall
Conditional Probability
Using Venn Diagram
Black
Ace
S
Event (Ace AND Black)
3 - 18
Black ‘Happens’:
Eliminates All
Other Outcomes
Black
(S)
Conditional Probability
Using Contingency Table
© 2003 Pearson Prentice Hall
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Type
P(Ace | Black) =
3 - 19
P(Ace AND Black)
P(Black)

2 / 52
26 / 52
Revised
Sample
Space

2
26
Statistical Independence
© 2003 Pearson Prentice Hall
1. Event Occurrence
Does Not Affect Probability of Another Event

P(A | B) = P(A)
Example: Toss 1 Coin Twice (independent)


P(second toss H)= ½
P(second toss H | first toss H) = ½
3 - 20
Tree Diagram
© 2003 Pearson Prentice Hall
Experiment: Select 2 Pens from 20 Pens:
14 Blue & 6 Red. Don’t Replace.
P(R) = 6/20
Dependent!
P(B) = 14/20
3 - 21
P(R|R) = 5/19
R
P(B|R) = 14/19
P(R|B) = 6/19
B
R
P(B|B) = 13/19
B
R
B
Thinking Challenge
© 2003 Pearson Prentice Hall
Using the Table Then the Formula, What’s
the Probability?
Pr(C)=
Event
C
D
4
2
P(B|C) =
Event
A
P(C|B) =
B
1
3
4
Total
5
5
10
Are C & B
Independent?
3 - 22
Total
6
Solution*
© 2003 Pearson Prentice Hall
Using the Formula, the Probabilities Are:
P(C  B) 1 / 10 1
P(C | B) =


P(B)
4 / 10 4
5 1
P(C) =

10 4
3 - 23
Dependent
© 2003 Pearson Prentice Hall
Multiplicative Rule
3 - 24
Multiplicative Rule
© 2003 Pearson Prentice Hall
1. Used to Get Compound Probabilities
for Intersection of Events

Called Joint Events
2. P(A and B) = P(A  B)
= P(A)*P(B|A)
= P(B)*P(A|B)
3. For Independent Events:
P(A and B) = P(A  B) = P(A)*P(B)
3 - 25
© 2003 Pearson Prentice Hall
Multiplicative Rule
Example
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
Type
Ace
P(Ace AND Black) = P(Ace)  P(Black | Ace)
 4  2
 2
       
 52   4 
 52 
3 - 26
Thinking Challenge
© 2003 Pearson Prentice Hall
Using the Multiplicative Rule, What’s the
Probability?
P(C  B) =
Event
C
D
4
2
P(B  D) =
Event
A
P(A  B) =
B
1
3
4
Total
5
5
10
3 - 27
Total
6
Solution*
© 2003 Pearson Prentice Hall
Using the Multiplicative Rule, the
Probabilities Are:
P(C  B) = P(C)  P(B|C) = 5/10 * 1/5 = 1/10
P(B  D) = P(B)  P(D|B) = 4/10 * 3/4 = 3/10
P(A  B) = P(A)  P(B|A)  0
3 - 28
Independence Revisited
© 2003 Pearson Prentice Hall
If A is independent of B, B is independent of A


P(A and B) = P(B|A)P(A)=P(A|B)P(B)
P(A|B)=P(A)  P(B|A)P(A) = P(A)P(B) P(B|A)=P(B)
Equivalence of the two independence definitions:
P(A and B) = P(A)*P(B) if and only if P(B|A) = P(B)



P(A and B) = P(A)P(B|A)
If P(B|A) = P(B), then P(A and B) = P(A)P(B)
If P(B|A) != P(B), then P(A and B) != P(A)P(B)
3 - 29
Random Variable
© 2003 Pearson Prentice Hall
3 - 30
Random Variables
© 2003 Pearson Prentice Hall
A random variable (rv) X is a mapping (function) from the
sample space S to the set of real numbers

If image(X ) finite or countable infinite, X is a discrete rv
Inverse image of a real number x is the set of all sample points
that are mapped by X into x:
It is easy to see that
3 - 31
Discrete Random Variable: pmf
© 2003 Pearson Prentice Hall
pk
3 - 32
Discrete Random Variable: CDF
1.2
© 2003 Pearson Prentice Hall
1
CDF
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
x
3 - 33
6
7
8
9
10
© 2003 Pearson Prentice Hall
Probability Mass Function
(pmf)
Ax : set of all sample points such that,
pmf
3 - 34
pmf Properties
© 2003 Pearson Prentice Hall
Since a discrete rv X takes a finite or a countably infinite set
values,
the last property above can be restated as,
3 - 35
Distribution Function
© 2003 Pearson Prentice Hall
pmf: defined for a specific rv value, i.e.,
Probability of a set

Cumulative Distribution Function (CDF)
3 - 36
© 2003 Pearson Prentice Hall
3 - 37
Distribution Function
properties
© 2003 Pearson Prentice Hall
Discrete Random
Variables
Equivalence:
Probability mass function
 Discrete density function
(consider integer valued random variable)

pk  P( X  k )
x
cdf:
F ( x) 
pmf:
pk  F (k )  F (k  1)
3 - 38

k 0
pk
© 2003 Pearson Prentice Hall
Common discrete random
variables
Constant
Uniform
Bernoulli
Binomial
Geometric
Poisson
Exponential
3 - 39
Discrete Random Vectors
© 2003 Pearson Prentice Hall
Examples:


Z=X+Y, (X and Y are random execution times)
Z = min(X, Y) or Z = max(X1, X2,…,Xk)
X:(X1, X2,…,Xk) is a k-dimensional rv defined on S

For each sample point s in S,
3 - 40
© 2003 Pearson Prentice Hall
3 - 41
Discrete Random Vectors
(properties)
Independent Discrete RVs
© 2003 Pearson Prentice Hall
X and Y are independent iff the joint pmf satisfies:
Mutual independence also implies:
Pair wise independence vs. set-wide independence
3 - 42
© 2003 Pearson Prentice Hall
Continuous Probability
Density Function
1.
Mathematical Formula
2.
Shows All Values, x, &
Frequencies, f(x)

3.
f(X) Is Not Probability
Frequency
(Value, Frequency)
f(x)
Properties
 f (x )dx  1
All X
(Area Under Curve)
f ( x )  0, a  x  b
3 - 43
a
b
Value
x
© 2003 Pearson Prentice Hall
Continuous Random
Variable Probability
d
Probability Is Area
Under Curve!
P (c  x  d)  c f ( x ) dx
f(x)
c
© 1984-1994 T/Maker Co.
3 - 44
d
X
© 2003 Pearson Prentice Hall
Normal Distribution
3 - 45
© 2003 Pearson Prentice Hall
Importance of
Normal Distribution
1. Describes Many Random Processes or
Continuous Phenomena
2. Can Be Used to Approximate Discrete
Probability Distributions

Example: Binomial
3. Basis for Classical Statistical Inference
3 - 46
Normal Distribution
© 2003 Pearson Prentice Hall
1.
‘Bell-Shaped’ &
Symmetrical
2.
Mean, Median,
Mode Are Equal
4.
Random Variable
Has Infinite Range
3 - 47
f(X)
X
Mean
Median
Mode
Probability
Density Function
© 2003 Pearson Prentice Hall
1
f ( x) 
e
 2
f(x)


x

3 - 48
=
=
=
=
=
 1  x    2



 2  

Frequency of Random Variable x
Population Standard Deviation
3.14159; e = 2.71828
Value of Random Variable (- < x < )
Population Mean
© 2003 Pearson Prentice Hall
Effect of Varying
Parameters ( & )
f(X)
B
A
C
X
3 - 49
© 2003 Pearson Prentice Hall
Normal Distribution
Probability
Probability is
area under
curve!
d
P(c  x  d )   f ( x) dx
c
f(x)
c
3 - 50
d
x
?
© 2003 Pearson Prentice Hall
Infinite Number
of Tables
Normal distributions differ by
mean & standard deviation.
f(X)
X
3 - 51
© 2003 Pearson Prentice Hall
Infinite Number
of Tables
Normal distributions differ by
mean & standard deviation.
Each distribution would
require its own table.
f(X)
X
That’s an infinite number!
3 - 52
© 2003 Pearson Prentice Hall
Normal Approximation
of Binomial Distribution
Mu = np
Sigma-squared = np(1-p)
Better approximation with
larger n

More on this when we
get to the central limit
theorem (chapter 6)
3 - 53
n = 10 p = 0.50
P(X)
.3
.2
.1
.0
0 2
X
4
6
8
10
© 2003 Pearson Prentice Hall
Inferential Statistics
3 - 54
Statistical Methods
© 2003 Pearson Prentice Hall
Statistical
Methods
Descriptive
Statistics
3 - 55
Inferential
Statistics
Inferential Statistics
© 2003 Pearson Prentice Hall
1.
Involves:


2.
Estimation
Hypothesis
Testing
Purpose

Make Inferences
about Population
Characteristics
3 - 56
Population?
Inference Process
© 2003 Pearson Prentice Hall
3 - 57
Inference Process
© 2003 Pearson Prentice Hall
Population
3 - 58
Inference Process
© 2003 Pearson Prentice Hall
Population
Sample
3 - 59
Inference Process
© 2003 Pearson Prentice Hall
Population
Sample
statistic
(X)
3 - 60
Sample
Inference Process
© 2003 Pearson Prentice Hall
Estimates
& tests
Sample
statistic
(X)
3 - 61
Population
Sample
Estimators
© 2003 Pearson Prentice Hall
1. Random Variables Used to Estimate a
Population Parameter

Sample Mean, Sample Proportion, Sample
Median
2. Example: Sample MeanX Is an
Estimator of Population Mean 

IfX = 3 then 3 Is the Estimate of 
3. Theoretical Basis Is Sampling Distribution
3 - 62
© 2003 Pearson Prentice Hall
Sampling Distributions
3 - 63
Sampling Distribution
© 2003 Pearson Prentice Hall
1. Theoretical Probability Distribution
2. Random Variable is Sample Statistic

Sample Mean, Sample Proportion etc.
3. Results from Drawing All Possible
Samples of a Fixed Size
4. List of All Possible [X, P(X) ] Pairs

Sampling Distribution of Mean
3 - 64
Expected Value of X-bar
© 2003 Pearson Prentice Hall
Remember “Useful Observation 1”
E(X+Y) = E(X) + E(Y)
Therefore
 X i  1
E X  E 
  E   X i 
 n  n
1
1
  E X i   n   
n
n
 
3 - 65
Variance of X-bar
© 2003 Pearson Prentice Hall
Remember Useful Obs. 3 for indep. X, Y

Var(X + Y) = Var(X) + Var (Y)
Therefore Var   X i   Var  X i 
Useful obs/exercise 4 Var (kX )  k 2Var  X 
 X i  1
Therefore
 
Var X Var 

1
 2 nVar  X i  
n
3 - 66
  2
n  n
Var  X i 
1
X 
X
n
n
Var  X 
i
© 2003 Pearson Prentice Hall
Properties of Sampling
Distribution of Mean
3 - 67
© 2003 Pearson Prentice Hall
Properties of Sampling
Distribution of Mean
1. Unbiasedness

Mean of Sampling Distribution Equals Population
Mean
2. Efficiency (minimum variance)

Sample Mean Comes Closer to Population Mean
Than Any Other Unbiased Estimator
3. Consistency

As Sample Size Increases, Variation of Sample
Mean from Population Mean Decreases
3 - 68
Unbiasedness
© 2003 Pearson Prentice Hall
P(X)
Unbiased
A
C

3 - 69
Biased
X
Efficiency
© 2003 Pearson Prentice Hall
P(X)
Sampling
distribution
of mean
B
Sampling
distribution
of median
A

3 - 70
X
Consistency
© 2003 Pearson Prentice Hall
P(X)
Larger
sample
size
B
Smaller
sample
size
A

3 - 71
X
© 2003 Pearson Prentice Hall
Sampling Distribution
Solution*
X   7.8  8
Z

  .50
 n 2 25
Sampling
Distribution
X   8.2  8
Z

 .50 Standardized
 n 2 25
Normal Distribution
X = .4
=1
.3830
.1915 .1915
7.8 8 8.2 X
3 - 72
-.50 0 .50
Z
© 2003 Pearson Prentice Hall
Sampling from
Normal Populations
3 - 73
© 2003 Pearson Prentice Hall
Sampling from
Normal Populations
Central Tendency
Population Distribution
= 10
x  
Dispersion

x 
n
Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =16
X = 2.5
X- = 50
3 - 74
X
X
© 2003 Pearson Prentice Hall
Sampling from
Non-Normal Populations
3 - 75
© 2003 Pearson Prentice Hall
Sampling from
Non-Normal Populations
Central Tendency
Population Distribution
= 10
x  
Dispersion

x 
n

Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =30
X = 1.8
X- = 50
3 - 76
X
X
© 2003 Pearson Prentice Hall
Central Limit Theorem
3 - 77
Central Limit Theorem
© 2003 Pearson Prentice Hall
3 - 78
Central Limit Theorem
© 2003 Pearson Prentice Hall
As
sample
size gets
large
enough
(n  30) ...
X
3 - 79
Central Limit Theorem
© 2003 Pearson Prentice Hall
As
sample
size gets
large
enough
(n  30) ...
sampling
distribution
becomes
almost
normal.
X
3 - 80
Central Limit Theorem
© 2003 Pearson Prentice Hall
As
sample
size gets
large
enough
(n  30) ...

x 
n
x  
3 - 81
sampling
distribution
becomes
almost
normal.
X
© 2003 Pearson Prentice Hall
Introduction
to Estimation
3 - 82
Statistical Methods
© 2003 Pearson Prentice Hall
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
3 - 83
Hypothesis
Testing
Estimation Process
© 2003 Pearson Prentice Hall
3 - 84
Estimation Process
© 2003 Pearson Prentice Hall
Population
Mean, , is
unknown

 
 


 
3 - 85
Estimation Process
© 2003 Pearson Prentice Hall
Population


Mean, , is
unknown

 
Sample


 
3 - 86
Random Sample
Mean 
X = 50
Estimation Process
© 2003 Pearson Prentice Hall
Population


Mean, , is
unknown

 
Sample


 
3 - 87
Random Sample
Mean 
X = 50
I am 95%
confident that
 is between
40 & 60.
Unknown Population
Parameters Are Estimated
© 2003 Pearson Prentice Hall
Estimate Population
Parameter...
Mean

Proportion
p
Variance

Differences
3 - 88
2
1 -  2
with Sample
Statistic
x
p^
s
2
x1 -x2
Estimation Methods
© 2003 Pearson Prentice Hall
3 - 89
Estimation Methods
© 2003 Pearson Prentice Hall
Estimation
3 - 90
Estimation Methods
© 2003 Pearson Prentice Hall
Estimation
Point
Estimation
3 - 91
Estimation Methods
© 2003 Pearson Prentice Hall
Estimation
Point
Estimation
3 - 92
Interval
Estimation
© 2003 Pearson Prentice Hall
Point Estimation
3 - 93
Point Estimation
© 2003 Pearson Prentice Hall
1. Provides Single Value

Based on Observations from 1 Sample
2. Gives No Information about How Close
Value Is to the Unknown Population
Parameter
3. Example: Sample MeanX = 3 Is Point
Estimate of Unknown Population Mean
3 - 94
© 2003 Pearson Prentice Hall
Interval Estimation
3 - 95
Estimation Methods
© 2003 Pearson Prentice Hall
Estimation
Point
Estimation
3 - 96
Interval
Estimation
Interval Estimation
© 2003 Pearson Prentice Hall
1. Provides Range of Values

Based on Observations from 1 Sample
2. Gives Information about Closeness to
Unknown Population Parameter

Stated in terms of Probability
3. Example: Unknown Population Mean Lies
Between 50 & 70 with 95% Confidence
3 - 97
© 2003 Pearson Prentice Hall
3 - 98
Key Elements of
Interval Estimation
© 2003 Pearson Prentice Hall
Key Elements of
Interval Estimation
Sample statistic
(point estimate)
3 - 99
© 2003 Pearson Prentice Hall
Key Elements of
Interval Estimation
Confidence
interval
Confidence
limit (lower)
3 - 100
Sample statistic
(point estimate)
Confidence
limit (upper)
© 2003 Pearson Prentice Hall
Key Elements of
Interval Estimation
A probability that the population parameter
falls somewhere within the interval.
Confidence
interval
Confidence
limit (lower)
3 - 101
Sample statistic
(point estimate)
Confidence
limit (upper)
© 2003 Pearson Prentice Hall
Confidence Limits
for Population Mean
We know the distribution of X-bar (for
large n:

CLT says it’s normally distributed with
mean Mu)

For any z, look up Pr   X  z  X
Equivalent formulations:
  X  zX
X   z  X ,   z  X


X  z X ,X  z X
3 -(102

zX


zX
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
3 - 103
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
x_

3 - 104
X
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
X =  ± Zx
x_

3 - 105
X
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
X =  ± Zx
x_
-1.65x

+1.65x
90% Samples
3 - 106
X
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
X =  ± Zx
x_
-1.65x
-1.96x

+1.65x
+1.96x
90% Samples
95% Samples
3 - 107
X
© 2003 Pearson Prentice Hall
Confidence Depends on
Interval (z)
X =  ± Zx
x_
-2.58x
-1.65x
-1.96x

+1.65x
+2.58x
+1.96x
90% Samples
95% Samples
99% Samples
3 - 108
X
Confidence Level
© 2003 Pearson Prentice Hall
1. Probability that the Unknown
Population Parameter Falls Within
Interval
2. Denoted (1 - 

 Is Probability That Parameter Is Not
Within Interval
3. Typical Values Are 99%, 95%, 90%
3 - 109
© 2003 Pearson Prentice Hall
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
/2
x = 
X
(1 - ) % of
intervals
contain .
Intervals
extend from
X - ZX to
X + ZX
3 - 110
_
 % do not.
Intervals derived from
many samples
© 2003 Pearson Prentice Hall
Factors Affecting
Interval Width
1. Data Dispersion

Measured by 
Intervals Extend from
X - ZX toX + ZX
2. Sample Size
—

X =  / n
3. Level of Confidence
(1 - )

Affects Z
© 1984-1994 T/Maker Co.
3 - 111
© 2003 Pearson Prentice Hall
3 - 112
Confidence Interval
Estimates
© 2003 Pearson Prentice Hall
Confidence Interval
Estimates
Confidence
Intervals
3 - 113
© 2003 Pearson Prentice Hall
Confidence Interval
Estimates
Confidence
Intervals
Mean
3 - 114
© 2003 Pearson Prentice Hall
Confidence Interval
Estimates
Confidence
Intervals
Mean
3 - 115
Proportion
© 2003 Pearson Prentice Hall
Confidence Interval
Estimates
Confidence
Intervals
Mean
3 - 116
Proportion
Variance
Confidence Interval
Estimates
© 2003 Pearson Prentice Hall
Confidence
Intervals
Mean
Known
3 - 117
Proportion
Variance
Confidence Interval
Estimates
© 2003 Pearson Prentice Hall
Confidence
Intervals
Mean
Known
3 - 118
Proportion
 Unknown
Variance
© 2003 Pearson Prentice Hall
Confidence Interval Estimate
Mean ( Known)
3 - 119
Confidence Interval
Estimates
© 2003 Pearson Prentice Hall
Confidence
Intervals
Mean
Known
3 - 120
Proportion
 Unknown
Variance
© 2003 Pearson Prentice Hall
Confidence Interval
Mean ( Known)
1. Assumptions



Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by
Normal Distribution (n  30)
3 - 121
© 2003 Pearson Prentice Hall

Confidence Interval
Mean ( Known)
1.Assumptions



Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by
Normal Distribution (n  30)
2. Confidence Interval Estimate
X  Z / 2 
3 - 122

n
   X  Z / 2 

n
© 2003 Pearson Prentice Hall
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence
interval estimate for  if  = 10.
3 - 123
© 2003 Pearson Prentice Hall
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence
interval estimate for  if  = 10.


X  Z / 2 
   X  Z / 2 
n
n
10
10
50  1.96 
   50  1.96 
25
25
46.08    53.92
3 - 124
© 2003 Pearson Prentice Hall
Confidence Interval Estimate
Mean ( Unknown)
3 - 125
Confidence Interval
Estimates
© 2003 Pearson Prentice Hall
Confidence
Intervals
Mean
Known
3 - 126
Proportion
 Unknown
Variance
Large Samples
© 2003 Pearson Prentice Hall
The sample variance s is a good estimator
of sigma
Carry on as before
3 - 127
© 2003 Pearson Prentice Hall
Another Way To Think
About It
Define variable
Z 
X 
X
X  X 


/ n s/ n
X-bar is the sampling distribution of the mean of a
sample of Xs
By the CLT, X-bar is normally distributed
Z is the normalized variable X

mu= 0 and sigma = 1
Confidence interval
find z-value associated with desired confidence level
alpha

De-normalize z-value to compute interval around X-bar
3 - 128

Problem for Small Samples
© 2003 Pearson Prentice Hall
X
s
n
may not be normally distributed
is not a good estimator of
3 - 129
X
© 2003 Pearson Prentice Hall
Solution for Small
Samples
1. Assumptions

Population of X Is Normally Distributed
2. Use Student’s t Distribution
X 
T 
s/ n
1.
Define variable
2.
T has the Student distribution with n-1 degrees of
freedom (When X is normally distributed)
There’s a different Student distribution for different
degrees of freedom
• As n gets large, Student distribution approximates a
normal distribution with mean = 0 and sigma = 1
3 - 130
•
Student’s t Distribution
© 2003 Pearson Prentice Hall
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
3 - 131
Z
t
© 2003 Pearson Prentice Hall
Confidence Interval
Mean ( Unknown)
Find t-value associated with desired
confidence level alpha Pr  T  t

 / 2 , n 1 
X 
T 
s/ n
100 1    confidence interval is
s
s 

, X  t  / 2 ,n 1
 X  t  / 2 ,n 1

n
n

3 - 132
Student’s t Table
© 2003 Pearson Prentice Hall
3 - 133
Student’s t Table
© 2003 Pearson Prentice Hall
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
3 - 134
Student’s t Table
© 2003 Pearson Prentice Hall
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
t values
3 - 135
Student’s t Table
© 2003 Pearson Prentice Hall
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
3 - 136
t
Student’s t Table
© 2003 Pearson Prentice Hall
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
3 - 137
t
Student’s t Table
© 2003 Pearson Prentice Hall
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
3 - 138
t
Student’s t Table
© 2003 Pearson Prentice Hall
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
.05
0
t values
3 - 139
t
Student’s t Table
© 2003 Pearson Prentice Hall
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
.05
0
t values
3 - 140
2.920
t
Degrees of Freedom (df)
© 2003 Pearson Prentice Hall
1. Number of Observations that Are Free
to Vary After Sample Statistic Has
Been Calculated
degrees of freedom
2. Example

Sum of 3 Numbers Is 6
X1 = 1 (or Any Number)
X2 = 2 (or Any Number)
X3 = 3 (Cannot Vary)
Sum = 6
3 - 141
= n -1
= 3 -1
=2
© 2003 Pearson Prentice Hall
Estimation Example
Mean ( Unknown)
A random sample of n = 25 hasx = 50 & s
= 8. Set up a 95% confidence interval
estimate for .
3 - 142
© 2003 Pearson Prentice Hall
Estimation Example
Mean ( Unknown)
A random sample of n = 25 hasx = 50 & s
= 8. Set up a 95% confidence interval
estimate for .
S
S
X  t  / 2, n 1 
   X  t  / 2, n 1 
n
n
8
8
50  2.0639 
   50  2.0639 
25
25
46.69    53.30
3 - 143
© 2003 Pearson Prentice Hall
Finding Sample Sizes
3 - 144
© 2003 Pearson Prentice Hall
(1)
Finding Sample Sizes
for Estimating 
Z
X 
x

Error
x
(2)
Error  Z x  Z
(3)
Z 
n
Error 2
2
2
Error Is Also Called Bound, B
3 - 145
I don’t want to
sample too much
or too little!

n
Determining Sample Size
© 2003 Pearson Prentice Hall
Z is determined by desired confidence
level
But how do you determine sigma?
3 - 146
Determining Sample Size
© 2003 Pearson Prentice Hall
Z is determined by desired confidence
level
But how do you determine sigma?



Known from previous studies
Pilot test on a small n
Theoretical derivation
3 - 147
Sample Size Example
© 2003 Pearson Prentice Hall
What sample size is needed to be 90%
confident of being correct within  5? A
pilot study suggested that the standard
deviation is 45.
3 - 148
Sample Size Example
© 2003 Pearson Prentice Hall
What sample size is needed to be 90%
confident of being correct within  5? A
pilot study suggested that the standard
deviation is 45.

Z
1.645 45
n

 219.2  220
2
2
Error
5
2
3 - 149
2
2
2
© 2003 Pearson Prentice Hall
End of Chapter
Any blank slides that follow are
blank intentionally.
3 - 150
Independent Discrete RVs
© 2003 Pearson Prentice Hall
X and Y are independent iff the joint pmf satisfies:
Mutual independence also implies:
Pair wise independence vs. set-wide independence
3 - 151
Discrete Convolution
© 2003 Pearson Prentice Hall
Let Z=X+Y . Then, if X and Y are independent,
In general, then,
3 - 152
Definitions
© 2003 Pearson Prentice Hall
Distribution function:
If FX(x) is a continuous function of x, then X is a
continuous random variable.


3 - 153
FX(x): discrete in x  Discrete rv’s
FX(x): piecewise continuous  Mixed rv’s
Definitions
© 2003 Pearson Prentice Hall
(Continued)
Equivalence:
CDF (cumulative distribution function)
PDF (probability distribution function)
Distribution function
FX(x) or FX(t) or F(t)
3 - 154
© 2003 Pearson Prentice Hall
Probability Density Function
(pdf)
X : continuous rv, then,
pdf properties:
1.
2.
3 - 155
Definitions
(Continued)
© 2003 Pearson Prentice Hall
Equivalence: pdf

probability density function

density function

density
dF
 f(t) =
dt
3 - 156
F (t )   f ( x )dx
t
 0 f ( x )dx
t
For a non-negative
random variable
Exponential Distribution
© 2003 Pearson Prentice Hall
Arises commonly in reliability & queuing theory.
A non-negative random variable
It exhibits memoryless (Markov) property.
Related to (the discrete) Poisson distribution


Interarrival time between two IP packets (or voice
calls)
Time to failure, time to repair etc.
Mathematically (CDF and pdf, respectively):
3 - 157
CDF of exponentially
distributed random variable
with  = 0.0001
© 2003 Pearson Prentice Hall
F(t)
12500
3 - 158
25000
t
37500
50000
Exponential Density Function
(pdf)
© 2003 Pearson Prentice Hall
f(t)
t
3 - 159