Unit i e-content

MATRUSRI ENGINEERING COLLEGE
DEPARTMENT OF ELECTRONICS COMMUNICATION
AND ENGINEERING
SUBJECT NAME: PROBABILITY THEORY & STOCHASTIC PROCESS
(PC201EC)-IIIrd SEMESTER
FACULTY NAME: Mr.A.ABHISHEK Reddy,Asst.Prof.
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PROBABILITY THEORY & STOCHASTIC PROCESS
COURSE OBJECTIVES:
1. To understand fundamentals of probability and Random variables as
applicable to Electronic Engg.
2. . To learn one Random Variable characteristic functions of different variables
using their density functions.
3. To learn Two Random Variable characteristic functions of different variables
4. To understand elementary concepts of stochastic process and their temporal
characteristics.
5. To understand elementary concepts of stochastic process and their spectral
characteristics.
COURSE OUTCOMES:
CO1:Understand different types of Random Variables, their distribution and
density functions.
CO2:Learn one Random variable characteristic functions of different variables
CO3:Extend the bi-variate distributions and the operations on them.
CO4:Understand elementary concepts of the Stochastic Processes in the
Temporal domain.
CO5:Analyze the frequency domain information of Stochastic Processes.
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SYLLABUS
UNIT I- Probability and Random Variable :
Concepts of Probability and Random Variable: Probability introduced
through Set Theory and Operations – Definitions and Axioms, Causality
versus Randomness, Borel Field, Probability Space – Discrete and Continuous,
Events - Definition and independent events, Joint Probability, Conditional
Probability, Repeated Trials, Combined Experiments, Bernoulli Trials,
Bernoulli’s Theorem, Total Probability, Baye’s Theorem.
Random Variable: Definition of a Random Variable, Conditions for a
Function to be random Variable, Discrete, Continuous and Mixed Random
Variables.
UNIT II- Distribution & Density Functions and Operations on One
Random Variable: Distribution and Density functions and their Properties -
Binomial, Poisson, Uniform, Gaussian, Gamma, Rayleigh and Conditional
Distribution, Methods of defining Conditional Event, Conditional Density,
Properties. Expected Value of a Random Variable, Function of a Random
Variable g(x) and its distribution, Moments about the Origin, Central
Moments, Variance and Skewness, Chebychev’s Inequality (no
proof),Characteristic Function, Moment Generating Function;
Transformations of Random Variables.
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UNIT III- Two Random Variables and operations: Bi-variate Distributions,
One Function of Two Random Variables, Two functions of two random
variables, Joint Distribution and Density Function and their properties, Joint
Moments, Joint Characteristic Functions, Conditional Distributions (Point &
Interval), Conditional Expected Values. Central Limit Theorem (no proof);
Engineering application (theoretical discussion) – Mutual information,
Channel Capacity and Channel Coding.
UNIT IV- Stochastic Processes – Temporal Characteristics: Introduction to
stationarity (First and Second order; WSS; SSS), statistical independence,
Time averages and ergodicity, random processes and independence, Mean-
Ergodic Processes, Correlation-Ergodic Processes, Autocorrelation Function
and its Properties, Cross-Correlation Function and its Properties, Covariance
and its Properties. Linear System Response of Mean and Mean-squared
Value. Introduction to Gaussian and Poisson Random Processes.
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UNIT V- Stochastic Processes – Spectral Characteristics: Power Spectral
Density and its properties; Relationship between Power Spectrum and
Autocorrelation Function; Relationship between Cross-Power Spectrum
and Cross-Correlation Function; White and colored noise, response to
linear systems and stochastic inputs, concept of Markov Processes.

TEXT BOOKS /REFERENCES
TEXT BOOKS:
1. Henry Stark and John W. Woods, Probability and Random
Processes with Application to Signal Processing, 3rd edition,
Pearson Education, 2014.
2. Athanasius Papoulis and S. Unnikrishna Pillai, Probability, Random
Variables and Stochastic Processes, 4th edition, McGraw Hill, 2006.
3. Peyton Z. Peebles, Probability, Random Variables & Random Signal
Principles, 4th edition, Tata McGraw Hill, 2001.
REFERENCES:
1. P. Ramesh Babu, Probability Theory and Random Process, 1st edition,
McGraw Hill Education (India) private limited.2015.
2. George R. Cooper and Clare D. McGillem “Probabilistic Methods of Signal
and System Analysis”,3rd edition, OXFORD UNIVERSITY PRESS
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LESSON PLAN:
UNIT I- Probability and Random Variable :
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S. No. Topic(S)
No.
of Hrs
Relevant
COs
Text Book/
Reference
Book
1. Probability introduced through Set
Theory and Operations
01 CO1 T3,R1
2. Borel Field, Probability Space – Discrete
and Continuous
01 CO1 T3,R1
3. Definitions and Axioms, Causality versus Randomness 01 CO1 T3,R1
4. Events - Definition and independent
events
01 CO1 T3,R1
5. Joint Probability, Conditional Probability 01 CO1 T3,R1
6. Repeated Trials, Combined
Experiments, Bernoulli Trials, Bernoulli’s Theorem
02 CO1 T3,R1
7. Total Probability, Bayes Theorem 02 CO1 T3,R1
8. Definition of a Random Variable,
Conditions for a Function to be a
Random Variable, Discrete, Continuous and Mixed
Random Variables.
01 CO1 T3,R1
TOTAL 10

PRE-REQUISITES FOR THIS COURSE: NIL
BUT KNOWLEDGE ON INTERGRATION,DIFFERENTIATION
AND TRIGONOMETRIC FORMULAE'S NEEDED.
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What is Probability?
Probability Theory, a branch of mathematics concerned with
the analysis of random phenomena. The outcome of a random
event cannot be determined before it occurs, but it may be any
one of several possible outcomes. The actual outcome is
considered to be determined by chance. The word probability and
its relatives(possible, probable, probably).

WHY SHOULD I STUDY?
1. Helps in developing critical thinking.
2. Helps students develop intuition into how the theory applies to
practical situations .
3. Teaches students how to apply probability theory to solving
engineering problems.
4. Prepare for COMPETITIVE EXAMS.
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The course PTSP forms a pre-requsite to :
1. Analog Communication.
2. Digital Communication.
3. Mobile and Cellular Communication.
4. Computer Network Theory. Etc…
The Actual definition of probability is given by “Relative
Frequency” and “Axiomatic” approach.

INTRODUCTION: Describes the basic principles of probability
through set theory and Axiomatic approach, also characterize
different types of Events and probability and apply to deal Total
probability, Baye’s Theorem, Bernoulli's Trials. Finally concept of
Random Variable is introduced and its types.
UNIT-I
OUTCOMES: After the completion of unit<students will be able to>
1. Explain the axiomatic approach of probability.
2. Quantify different types of events and different probability
definitions.
3. Apply joint and conditional probability to understand Total
probability, Bayes Theorem & Bernoulli's trials.
4. Understand the concept of random variable and its types.
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CONTENTS:
1. BASIC CONCEPTS OF SET THEORY.
2. VENN DIAGRAM
3.SET OPERATIONS
4. PROPERTIES OF SET OPERATIONS.
OUTCOMES: After the completion of Module<students will be able to>
Idealize the concepts of Set Theory and its properties.
MODULE-I
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THE MODERN APPROACH TO PROBABILITY BASED ON AXIOMATICALLY DEFINING
PROBABILITY AS FUNCTION OF A SET.
A BACKGROUND ON THE SET THEORY IS ESSENTIAL FOR UNDERSTANDING
PROBABILITY
SOME OF THE BASIC CONCEPTS OF SET THEORY ARE INTRODUCED HERE.
Set:
A SET IS A “WELL DEFINED” COLLECTION OF OBJECTS. THESE OBJECTS ARE
CALLED ELEMENTS OR MEMBERS OF THE SET. USUALLY UPPERCASE LETTERS ARE
USED TO DENOTE SETS.
Example 1
A={a,e,i,o,u} IS A SET .
BASIC CONCEPTS OF SET THEORY
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EXAMPLE-2
“the set of former Nobel prize winners”
-- well-defined
EXAMPLE-3:
“the set of tall students in our college”
---> not well-defined
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METHODS FOR DEFINING SET:
1. TABULAR METHOD: The Elements Are Enumerated Explicitly.
EX: set of all integer's b/w 5 and 10 be
Z={ 6,7,8,9}
2. RULE METHOD: The Content Is Determined By Some Rule.
EX : Z= { z/ intergers b/w 5 and 10}
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CARDINAL NUMBER OF SET: The number of distinct elements in a given
set, say “A” is called cardinal number of set. It is denoted by n(A).
Ex: A={ 1,2,3,4} => n(A)=4
EQUIVALENT SETS: Two sets A and B are said be equal if their cardinal
number is same i.e n(A)=n(B).
Ex: A={1,2,3,4) => n(A)=4
B={p,q,r,s} => n(B)=4 therefore, A< -- > B
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Venn diagrams were developed by the logician john venn (1834 – 1923). In
these diagrams, the universal set is represented by a rectangle and other sets
of interest within the universal set are depicted as closed –plane figures.
The rectangle represents the universal set, U, while the portion bounded
by the circle represents set A.
VENN-Diagram
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U
A

COMPLEMENT OF A SET
The colored region inside U and outside the circle is labeled A' (read “A
prime”). This set, called the complement of A, contains all elements that
are contained in U but not in A.
For any set A within the universal set U, the complement of A, written A',
is the set of all elements of U that are not elements of A. That is
VENN-Diagram
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U
A
A
{ | and }.
A x x U x A
   

Subsets of a set : set A is a subset of set B if every element of A is also an
element of B. In symbols this is written
Set A is a proper subset of set B if written as
VENN-Diagram
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.
A B

U
A
B
and .
A B
 .
A B


SET UNION: “A union B” is the set of all elements that are in A, or B, or
both.
This is similar to the logical “or” operator
SET INTERSECTION: “A intersect B” is the set of all elements that are in
both A and B.
This is similar to the logical “and”
The intersection of A and B is: A ∩ B = {x|x ∈ A and x ∈ B}
SET OPERATIONS
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A B

The union of A and B is: A ∪ B = {x|x ∈ A or x ∈ B}
A B


SET DIFFERENCE: The set difference “A minus B” is the set of elements
that are in A, with those that are in B subtracted out. Another way of
putting it is, it is the set of elements that are in A, and not in B.
THE DIFFERENCE OF A AND B IS: A - B = {X|X ∈ A AND X / ∈ B}
SET OPERATIONS
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A B A B
  
A
B

1. IDENTITY PROPERTY:
2. COMMUTATIVE PROPERTY:
3. ASSOCIATIVE PROPERTY:
4 DISTRIBUTIVE PROPERTY:
Properties of SET Operations
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5.COMPLEMENTARY PROPERTY:
6.DE-MORGANS LAWS: FOR ANY SETS A AND B
Properties of SET Operations
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7.POWER SET: P(A) is the set of all possible subsets of the set A
The set of all the subsets of a set S is called the power set of S and
denoted by P(A)
EX:

CONTENTS:
1. BASIC CONCEPTS?TERMINOLOGIES
2. DEFINITIONS OF PROBABILITY.
3. BASIC RULES OF PROBABILITY
Define probability based on Axiomatic approach.
MODULE-II
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Experiment: An experiment consists of procedure, observation and
model.
(Or)
Any procedure performed infinite number of times and has well
defined set of possible outcomes.
Example-1.1
Procedure: Flip a coin and let it land on a table.
Observation: Observe which side (head or tail) faces you after the coin lands.
Model: Heads and tails are equally likely. The result of each flip is unrelated to
the results of previous flips.
Example 1.2 Flip a coin three times. Observe the sequence of
heads and tails.
Example 1.3 Flip a coin three times. Observe the number of heads.
These two experiments have the same procedure different but
have different observations(called ” OUTCOMES”)
BASIC CONCEPTS/TERMINOLOGY
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RANDOM EXPERIMENT : An experiment is a random experiment if its
outcome cannot be predicted precisely. One out of a number of
outcomes is possible in a random experiment.
OUTCOME: Any possible observation of an experiment.
TRAIL: A single performance of the experiment.
SAMPLE SPACE: The set of all possible outcomes in any given experiment.
(OR)
It is the finest-grain, mutually exclusive, collectively exhaustive set of all
possible outcomes of an experiment.
In General Sample space is denoted by “S”
• A sample space may be finite , countably infinite or uncountable.
• A finite or countably infinite sample space is called a discrete sample
space.
• An uncountable sample space is called a continuous sample space
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COMPARISON:
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SET THEORY PROBABILITY
SET EVENT
UNIVERSAL SET SAMPLE SPACE
ELEMENTS OUTCOMES

EXAMPLE 2 THROWING A FAIR DIE:
THE ASSOCIATED SAMPLE SPACE S={1,2,3,4,5,6}
Example 3 Tossing a fair coin until a head is obtained
We may have to toss the coin any number of times before a head is
obtained. Thus the possible outcomes are :
H, TH,TTH,TTTH, …..
How many outcomes are there? The outcomes are countable but infinite
in number. So its is Countably-infinite sample space.
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EXAMPLE 4 :
PICKING A REAL NUMBER AT RANDOM BETWEEN -1 AND +1
THE ASSOCIATED SAMPLE SPACE IS
CLEARLY “S “ IS CONTINUOUS SAMPLE SPACE.
EXAMPLE 5 : OUTPUT OF A RADIO RECEIVER AT ANY TIME
Suppose the output voltage of a radio receiver at any time t is a value
lying between -5 V and 5V.
CLEARLY “S “ IS CONTINUOUS SAMPLE SPACE.
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CERTAIN & IMPOSSIBLE EVENTS:
An event is certain if it always appears as an outcome.
An event which never occurs is an impossible event.
MUTUALLY EXCLUSIVE EVENTS(MEE):
If any two events in an experiment have no common outcomes then
the events are said to be mutually exclusive events.
Two events are mutually exclusive if, when one event occurs,
the other cannot, and vice versa.
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EQUALLY LIKELY EVENTS(ELE): If one of the events cannot be expected to
happen in preference to another, then such events are said to be equally likely
events.
( Or) Each outcome of the random experiment has an equal chance of
occurring.
(OR) Two or more events which have an equally likely chance or equal
probability of occurrence are said to be equally likely. I.E if on taking into
account all the conditions , there should be no reason to except any one of the
events in preference over the others.
Equally likely Events may be mutually exclusive or not mutually exclusive
(or) Vive-versa.
Case(i): ELE & NOT MEE
Case(ii): ELE & MEE
Case(iii): NOT ELE & MEE
Case(iv): NOT ELE & NOT MEE
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RELATIVE-FREQUENCY BASED DEFINITION OF PROBABILITY:
Suppose that a random experiment repeated “n” times and if the event
A occurs n(A) times, then the probability of event A is defined as the
relative frequency of event A when the number of trials n tends to
infinity.
Mathematically,
where n(A) is the number of times the Event A has occurred and n is
the number of trials.
Example: Suppose a die is rolled 500 times. The following table shows
the frequency each face.
PROBABILITY DEFINITIONS
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Face 1 2 3 4 5 6
frequency 82 81 88 81 90 78
Relative freq. 0.164 0.162 0.176 0.162 0.18 0.156

We see that the relative frequencies are close to 1/6. How do we
ascertain that these relative frequencies will approach to 1/6 , as we
repeat the experiments infinite no of times?
Discussion: This definition is also inadequate from the theoretical
point of view.
(i)We cannot repeat an experiment infinite number of times.
(ii) How do we ascertain that the above ratio will converge for all
possible sequences of outcomes of the experiment?
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DISCUSSION
The classical definition is limited to a random experiment which has
only a finite number of outcomes. In many experiments like that in the
above examples, the sample space is finite and each outcome may be
assumed ‘equally likely.' In such cases, the counting method can be
used to compute probabilities of events.
Consider the experiment of tossing a fair coin until a ‘head' appears. As
we have discussed earlier, there are countably infinite outcomes. Can
you believe that all these outcomes are equally likely?
The notion of equally likely is important here. Equally likely means
equally probable. Thus this definition presupposes that all events
occur with equal probability . Thus the definition includes a concept to
be defined.
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BASIC RULES OF PROBABILITY
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4. If
we have,
we can similarly show that
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5. If
we have,
6. We can apply the properties of sets to establish the following result for
The following generalization is known as the principle inclusion-exclusion.
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PRINCIPLE OF INCLUSION-EXCLUSION:
Suppose,
Then,
PROBABILITY ASSIGNMENT IN A DISCRETE SAMPLE SPACE:
Consider a finite sample space . then the sigma algebra is defined by the power
set of S. For any elementary event , we can assign a probability P( Si) such that,
For any Event, A belongs to F , we can define the probability as,
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In a special case, when the outcomes are equi-probable, we can assign equal
probability p to each elementary event.
Ex-1: Consider the experiment of rolling a fair die
Suppose represent the elementary events. Thus,a1 is the event getting
1,A2 is the event getting 2 and so on…. . Since all six disjoint events are
equiprobable and we get
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Suppose A is event getting and Odd face then,
Ex-2: Consider the experiment of tossing a fair coin until a head is obtained.
Here S={H,TH,TTH,……….}. Let us call
And So on. If we assign, ,then
Let A=(S1,S2,S3) is the event of obtaining the head before the 4 th toss. Then
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CONTENTS:
PROBABILITY USING COUNTING METHOD
Learn basic fundamentals related to permutations and combinations.
MODULE-III
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In many applications we have to deal with a finite sample space S and the
elementary events formed by single elements of the set may be assumed
equiprobable. In this case, we can define the probability of the
event A according to the classical definition discussed earlier:
P(A)=n(A)/n
where n(A) is the number of elements favourable to event A and n is the total
number of elements in the sample space S.
Thus calculation of probability involves finding the number of elements in the
sample space S and the event A . combinatorial rules give us quick algebraic
formulae to find the elements in S. We briefly outline some of these rules:
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The plot of probability vs number of students is shown in above table. Observe the steep
rise in the probability in the beginning. In fact this probability for a group of 25 students is
greater than 0.5 and that for 60 students onward is closed to 1. This probability for 366 or
more number of students is exactly one.
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CONTENTS:
1. JOINT PROBABILITY
2. CONDITIONAL PROBABILITY
Estimate the joint and conditionalness for a given experiment.
MODULE-IV
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JOINT PROBABILITY
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CONDITIONAL PROBABILITY
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The table below shows the number of survey subjects who have received and not
received a speeding ticket in the last year, and the color of their car. Find the
probability that a randomly chosen person:
a) has a speeding ticket given they have a red car
b) has a red car given they have a speeding ticket
CONDITIONAL PROBABILITY
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CONTENTS:
1. TOTAL PROBABILITY
2. BAYES PROBABILITY
Understand the importance of Total probability and Bayes theorems.
MODULE-V
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TOTAL PROBABILITY
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TOTAL PROBABILITY
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A
B1
B3
B2
Bn
B4
Bm
B1 S

BAYE’S THEOREM
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From the total probability theorem,
or,
BAYE’S THEOREM
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In Baye’s theorem, the probabilities P(Bn) are usually referred as
“ priori probabilities”, since they apply to the events Bn before the
performance of the experiment.
The conditional probabilities sometimes called as “ transistion
probabilities”, and
P(Bn/A) are called “posteriori probabilities”, since they apply after
the experiment performance when some event A is obtained.
BAYE’S THEOREM
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CONTENTS:
1.INDEPENDENT EVENTS
2. REPEATED TRIALS
Interpret Independent Events and repeated trails.
MODULE-VI
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INDEPENDENT EVENTS
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In our discussions so far, we considered the probability defined over a sample
space corresponding to a random experiment. Often, we have to consider several
random experiments in a sequence. For example, the experiment corresponding
to sequential transmission of bits through a communication system may be
considered as a sequence of experiments each representing transmission of single
bit through the channel.
A combined experiment consists of forming a single experiment by suitably
combining individual experiment, which we call subexperiments.
Combined Sample Space: Consider two subexperiments and let S1 and S2 be
sample spaces and let s1 and s2 represent elements. The combined sample space
S whose elements are all ordered pairs (s1,s2).
S=S1*S2
For N sample spaces, Sn, n=1,2,3……,N having elements Sn then
S=S1*S2*S3*…...............*SN
REPEATED TRIALS
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REPEATED TRIALS
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CONTENTS:
1.BERNOULLI’S TRIALS
2. BERNOULLI’S THEOREM
Understand Bernoulli’s Theorem
MODULE-VII
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BERNOULLI’S TRIALS
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BERNOULLI’S THEOREM
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CONTENTS:
1.RANDOM VARIABLE
2. TYPES OF RV
Understand Random Variable and its types.
MODULE-VIII
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A Random Variable is a function which maps all the events(outcomes)
in a sample space on to a real line. A random variable is represented by
a capital letter(such as W,X,Y or Z) and any particular value of the
random variable by lower case letter(such as w,x,y,z or z)
RANDOM VARIABLE
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Ex: Consider an experiment consists of rolling a die and flipping a coin.
Let a RV be a function of X that (i) a coin head and number
corresponds to positive value (ii) a coin Tail and number corresponds
to negative value .
RANDOM VARIABLE
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RANDOM VARIABLE
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RANDOM VARIABLE
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Experiment Outcome Random Variable Range of
Random
Variable
Stock 50
Xmas trees
Number of
trees sold
X = number of
trees sold
0,1,2,, 50
Inspect 600
items
Number
acceptable
Y = number
acceptable
0,1,2,…,
600
Send out
5,000 sales
letters
Number of
peoplee
responding
Z = number of
people responding
0,1,2,…,
5,000
Build an
apartment
building
%completed
after 4
months
R = %completed
after 4 months
0 R100
Test the
lifetime of a
light bulb
(minutes)
Time bulb
lasts - up to
80,000
minutes
S = time bulb
burns
0 S 80,000

TYPES RANDOM VARIABLE
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1. A biased coin is tossed till a head appears for the first time. What is the
probability that the number of required tosses is odd?
2. Three horses A, B and C are in a race. A is twice as likely to win as B and B is
twice likely to win as C. What is the probability that B or C wins?
3. A noisy transmission channel has a per-digit error probability of 0.01. Calculate
the probability of more than one error in 10 received digits.
4. State the fundamental axioms of probability.
5. A box contains 50 diodes of which 1 0 are known to be bad. A diode is selected
at random.
(a) What is the probability that it is bad?
(b) If the first diode drawn from the box was good, what is the probability that a
second diode drawn will be good?
(c) If two diodes are drawn from the box what is the probability that they are
both good?
ASSIGNMNET QUESTIONS
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6. A manufacturer of electronic equipment purchases 1 000 ICs from supplier A," 2000 ICs
from supplier 8, and 3000 ICs from supplier C. Testing reveals that the conditional probability
of an IC failing during burn-in is, for devices from each of the suppliers P(F/A) = 0.05, P(F/B) =
0. 10, P (F/C) = 0.10. The ICs from all suppliers are mixed together and one device is selected
at random.
a) What is the probability that it will fail during. burn-in?
b} Given that the device fails, what is the probability that the device came from supplier A?
7. A manufacturer of electronic equipment buys 1 000 ICs for which the probability of one IC
being bad is 0.01 . Determine
a) What is the probability that exactly 1 O of the ICs are bad?
b) What is the probability that none of the ICs is bad?
c) What is the probability that exactly one of the ICs is bad?
8. In the experiment of throwing two fair dice, let A be the event that the first die is odd, B
be the event that the second die is odd, and C be the event that the sum is odd. Show that
events A, B, and C are pairwise independent, but A, B, and C are not independent.
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9. There are two identical decks of cards, each possessing a distinct symbol so that the cards
from each deck can be identified. One deck of cards is laid out in a fixed order, and the other
deck is shuffled and the cards laid out one by one on top of the fixed deck. Whenever two
cards with the same symbol occur in the same position, we say that a match has occurred.
Let the number of cards in the deck be 10. Find the probability of getting a match at the first
four positions.
10. It is estimated that 50% of emails are spam emails. Some software has been
applied to filter these spam emails before they reach your inbox. A certain brand of software
claims that it can detect 99% of spam emails, and the probability for a false positive (a non-
spam email detected as spam) is 5%. Now if an email is detected as spam, then what is the
probability that it is in fact a non-spam email?
MATRUSRI
ENGINEERING COLLEGE

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