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PROBABILITY AND STATISTICS
FOR ENGINEERING
Hossein Sameti
Sharif University of Technology
Spring 2008
Text Book
A. Papoulis and S. Pillai,
Probability, Random Variables and Stochastic Processes,
4th Edition, McGraw Hill, 2000
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TABLE OF CONTENTS
PROBABILITY THEORY
Lecture – 1
Lecture – 2
Lecture – 3
Lecture – 4
Lecture – 5
Lecture – 6
Lecture – 7
Lecture – 8
Lecture – 9
Lecture – 10
Lecture – 11
Lecture – 12
Lecture – 13
Basics
Independence and Bernoulli Trials
Random Variables
Binomial Random Variable Applications, Conditional
Probability Density Function and Stirling’s Formula.
Function of a Random Variable
Mean, Variance, Moments and Characteristic Functions
Two Random Variables
One Function of Two Random Variables
Two Functions of Two Random Variables
Joint Moments and Joint Characteristic Functions
Conditional Density Functions and Conditional Expected Values
Principles of Parameter Estimation
The Weak Law and the Strong Law of Large numbers
STOCHASTIC PROCESSES
Lecture – 14
Lecture – 15
Lecture – 16
Lecture – 17
Lecture – 18
Lecture – 19
Lecture – 20
Stochastic Processes - Introduction
Poisson Processes
Mean square Estimation
Long Term Trends and Hurst Phenomena
Power Spectrum
Series Representation of Stochastic processes
Extinction Probability for Queues and Martingales
Source AT: www.mhhe.com/papoulis
3
Teaching Assistants:
Hoda Akbari
Course Evaluation:
Homeworks: 10-20%
Midterm: 25-30%
Final Exam: 50-60%
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Probability And Statistics
Source: http://ocw.mit.edu
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PROBABILITY THEORY
1.Basics
Random Phenomena, Experiments
 Study of random phenomena
 Different outcomes
 Outcomes that have certain underlying patterns about them
 Experiment
- repeatable conditions
 Certain elementary events Ei occur in different but completely
uncertain ways.
 probability of the event Ei : P(Ei )>=0
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Probability Definitions
 Laplace’s Classical Definition
- without actual experimentation
- provided all these outcomes are equally likely.
Example
• a box with n white and m red balls
elementary outcomes: {white , red}
Probability of “selecting a white ball”:
• P a given number is divisible by a prime p 
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p
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Probability Definitions
 Relative Frequency Definition
- The probability of an event A is defined as
- nA is the number of occurrences of A
- n is the total number of trials
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Probability Definitions
Example
1. The probability that a given number is divisible by a prime p:
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Counting - Remark
 General Product Rule
if an operation consists of k steps each of which can be performed in ni ways
(i = 1, 2, …, k), then the entire operation can be performed in ni ways.
Example
- Number of PINs
- Number of elements in a Cartesian product
- Number of PINs without repetition
- Number of Input/Output tables for a circuit with n input signals
- Number of iterations in nested loops
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Permutations and Combinations - Remark
 If order matters choose k from n:
- Permutations :
 If order doesn't matters choose k from n:
- Combinations :
Example
A fair coin is tossed 7 times. What is the
probability of obtaining 3 heads? What is
the probability of obtaining at most 3
heads?
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Example: The Birthday Problem
 Suppose you have a class of 23 students. Would you think it likely or
unlikely that at least two students will have the same birthday?
 It turns out that the probability of at least two of 23 people having the
same birthday is about 0.5 (50%).
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Axioms of Probability- Basics
 The axiomatic approach to probability, due to Kolmogorov,
developed through a set of axioms
 The totality of all events known a priori, constitutes a set Ω, the set
of all experimental outcomes.
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Axioms of Probability- Basics
 A and B are subsets of Ω .
A
A
B
A B
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B
A
A B
A
A
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Mutually Exclusiveness and Partitions
A  B  ,
 A and B are said to be mutually exclusive if
 A partition of  is a collection of mutually exclusive(ME) subsets of
 such that their union is .
A1
A
B
Aj
A2
Ai
An
A B  
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De-Morgan’s Laws
A B  A B ;
A
B
A B
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A
B
A B
A B  A B
A
B
A
B
A B
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Events
 Often it is meaningful to talk about at least some of the subsets of 
as events
 we must have mechanism to compute their probabilities.
Example
Tossing two coins simultaneously:
A: The event of “Head has occurred at least once” .
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Events and Set Operators
 “Does an outcome belong to A or B”
 “Does an outcome belong to A and B”
 “Does an outcome fall outside A”?
 These sets also qualify as events.
 We shall formalize this using the notion of a Field.
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Fields
 A collection of subsets of a nonempty set  forms a field F if
(i)
F
(ii)
If A  F , then A  F
(iii) If A  F and B  F , then A  B  F .
 Using (i) - (iii), it is easy to show that the following also belong to F.
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Fields
If
then
 We shall reserve the term event only to members of F.
 Assuming that the probability P(Ei ) of elementary outcomes Ei of Ω are
apriori defined.
 The three axioms of probability defined below can be used to assign
probabilities to more ‘complicated’ events.
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Axioms of Probability
 For any event A, we assign a number P(A), called the probability of
the event A.
 Conclusions:
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Probability of Union of to Non-ME Sets
A
AB
A B
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Union of Events
 Is Union of denumerably infinite collection of
pairwise disjoint events Ai an event?
 If so, what is P(A ) ?
 We cannot use third probability axiom to
compute P(A), since it only deals with two
(or a finite number) of M.E. events.
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An Example for Intuitive Understanding
 in an experiment, where the same coin is tossed indefinitely define:
A = “head eventually appears”.
 Our intuitive experience surely tells us that A is an event.
If
An  head appears for the 1st time on the nth toss
 {t
,
t, 
t ,
, t , h}


n 1
We have:
 Extension of previous notions must be done based on our intuition as
new axioms.
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σ-Field (Definition):
 A field F is a σ-field if in addition to the
three mentioned conditions, we have the
following:
- For every sequence of pairwise disjoint
events belonging to F, their union also
belongs to F
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Extending the Axioms of Probability
 If Ai s are pairwise mutually exclusive
 from experience we know that if we keep
tossing a coin, eventually, a head must
show up:
 But:
P ( A)  1.
A

A
,
 P( A
).
n
n 1
 Using the fourth probability axiom we
have:



P( A)  P
  An 

 n 1

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
n 1
n
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Reasonablity
 In previously mentioned coin tossing experiment:
 So the fourth axiom seems reasonable.
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Summary: Probability Models
 The triplet (, F, P)
-  is a nonempty set of elementary events
-F
is a -field of subsets of .
- P is a probability measure on the sets in F subject the four axioms
 The probability of more complicated events must follow this
framework by deduction.
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Conditional Probability
 In N independent trials, suppose NA, NB, NAB denote the number of times
events A, B and AB occur respectively.
 According to the frequency interpretation of probability, for large N,
P( A) 
NA
N
N
, P( B )  B , P( AB)  AB .
N
N
N
 Among the NA occurrences of A, only NAB of them are also found among the
NB occurrences of B.
 Thus the following is a measure of “the event A given that B has already
occurred”:
N AB N AB / N P( AB)


NB
NB / N
P( B )
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Satisfying Probability Axioms
 We represent this measure by P(A|B) and define:
P( AB)
P( A | B ) 
,
P( B )
P( B )  0.
 As we will show, the above definition is a valid one as it satisfies all
probability axioms discussed earlier.
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Satisfying Probability Axioms
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Properties of Conditional Probability
Example
In a dice tossing experiment,
-
A : outcome is even
-
B: outcome is 2.
The statement that B has occurred
makes the odds for A greater
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Law of Total Probability
 We can use the conditional probability to express the probability of a
complicated event in terms of “simpler” related events.
 Suppose that
 So,
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Conditional Probability and Independence
 A and B are said to be independent events, if
P( AB)  P( A)  P( B ).
 This definition is a probabilistic statement, not a set theoretic notion
such as mutually exclusiveness.
 If A and B are independent,
P( A | B ) 
P( AB)
P( A) P( B )

 P( A).
P( B )
P( B )
 Thus knowing that the event B has occurred does not shed any
more light into the event A.
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Independence - Example
Example
From a box containing 6 white and 4 black balls, we remove two balls at
random without replacement.
What is the probability that the first one is white and the second one is black?
P(W1  B2 )  ?
W1  B2  W1B2  B2W1.
P(W1B2 )  P( B2W1 )  P( B2 | W1 ) P(W1 ).
P (W1 ) 
6
6
3

 ,
6  4 10 5
P( B2 | W1 ) 
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P(W1 B2 ) 
4
4
 ,
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3 4 12
 
5 9 45
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Example - continued
 Are W1 and B2 independent?
Removing the first ball has two possible outcomes:
These outcomes form a partition because:
So,
P( B2 )  P( B2 | W1 ) P(W1 )  P( B2 | B1 ) P( B1 )

4 3
3
4 4 3 1 2 42 2
 

    
 ,
5  4 5 6  3 10 9 5 3 5
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Thus the two events are not independent.
P( B2 ) P(W1 ) 
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2 3
4
  P( B2W1 ) 
5 5
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General Definition of Independence
 Independence between 2 or more events:
 Events A1,A2, ..., An are mutually independent if, for all possible
subcollections of k ≤ n events:
Example
In experiment of rolling a die,
A = {2, 4, 6}
B = {1, 2, 3, 4}
C = {1, 2, 4}.
Are events A and B independent?
What about A and C?
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Source: http://ocw.mit.edu
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Bayes’ Theorem
We have:
P( A | B ) 
Thus,
P( AB)
,
P( B )
P( AB)  P( A | B ) P( B ).
Also,
P( B | A) 
P( BA)
P( AB)

,
P( A)
P( A)
P( AB)  P( B | A) P( A).
P ( A | B ) P ( B )  P ( B | A) P ( A).
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Bayesian Updating: Application Of Bayes’ Theorem
 Suppose that A and B are dependent events and A has apriori
probability of P(A ) .
 How does Knowing that B has occurred affect the probability of A?
 The new probability can be computed based on Bayes’ Theorm.
 Bayes’ Theorm shows how to incorporate the knowledege about B’s
occuring to calculate the new probability of A.
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Bayesian Updating - Example
Example
 Suppose there is a new music device in the market that plays a new
digital format called MP∞. Since it’s new, it’s not 100% reliable.
 You know that
- 20% of the new devices don’t work at all,
- 30% last only for 1 year,
- and the rest last for 5 years.
 If you buy one and it works fine, what is the probability that it will last for 5
years?
Source: http://ocw.mit.edu
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Generalization of Bayes’ Theorem
A more general version of Bayes’ theorem involves partition of Ω :
P ( Ai | B ) 
P ( B | Ai ) P ( Ai )

P( B )
P ( B | Ai ) P ( Ai )
n
 P( B | A ) P( A )
i 1
In which,
i
,
i
Ai , i  1  n,
Represents a collection of mutually exclusive events with assiciated apriori
probabilities:
P( Ai ), i  1  n.
With the new information “B has occurred”, the information about Ai can be
updated by the n conditional probabilities:
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Bayes’ Theorem - Example
Example
1. Two boxes, B1 and B2 contain 100 and 200 light bulbs
respectively. The first box has 15 defective bulbs and the second 5.
Suppose a box is selected at random and one bulb is picked out.
a) What is the probability that it is defective?
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Example - Continued
Suppose we test the bulb and it is found to be defective. What is the
probability that it came from box 1?
P( B1 | D)  ?
P( B1 | D) 
P( D | B1 ) P( B1 ) 0.15  1 / 2

 0.8571.
P( D)
0.0875
 Note that initially,
P( B1 )  0.5;
 But because of greater ratio of defective bulbs in B1 ,this probability
is increased after the bulb determined to be defective..
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