This document provides details about a course on random variables and stochastic processes. It includes: - An overview of the course content which will cover probability theory, random variables, distributions, and stochastic processes. - Information about assignments, quizzes, grading policy, textbooks, and the instructor's office hours. - Examples and explanations of key concepts from probability theory that will be covered, including sample spaces, probability values, events, and complements of events. Applications to games of chance, software errors, and power plant operations are discussed. - The goal of developing mathematical tools to analyze and characterize random signals and stochastic processes is stated.

1. Random Variable and
Stochastic Processes
Instructor: Muhammad Shahid
Iqbal
(m.shahid@uog.edu.pk)

2. Message
• Have a deep trust in Almighty Allah He loves
you. Always bow in front of Him. You will be
successful.
• Defeat is not final until you stop trying.
• I am not in a competition to anybody but my
self. My target is to beat my last performance.
• Always be honest, its key to success.

3. Course Details
Course Code EE-
Credit Hours 3(3+0)
Title Random Variables and Stochastic processes
Instructor Muhammad Shahid Iqbal
Office hours: Tuesday 3:00-4:30, Thursday 12:00-1:30
Office and Consulting Timings Room Number 207 EE Department
Grading Policy
Sessional Marks: 25 Marks (detail as under)
Quizes:5 Marks of sessional
Assignments:10 Marks of sessional
Research Paper Presentation 10 Marks
Mid term Paper: 25%
Final Paper: 50%
Text/Reference Books Probability, Random Variables and
random Signal principles. By P.Z.Peebles.
4th
Edition

4. Assignments, Quizzes and Attendance
40% marks will be deducted on late
submission of assignment (only one day
Late). After this no assignment will be
submitted.
No compensation for announced missed
quiz, Assignment (what ever the reason) will
be offered so please do not waste your time
on argues.
Keep the record of your attendance no one
will be facilitated in this regard.

5. Paper
• What ever we have studied in the Class and its
application will be the part of assessment.
• All Home take and submitted assignments.
• End exercise of the text book.
• Any thing else discussed in future.

6. Course
Set theory, basic concepts of probability,
conditional probability, independent events,
Baye's formula, discrete and continuous
random variables, distributions and density
functions, probability distributions (binomial,
Poisson, hyper geometric, normal, uniform and
exponential),mean, variance, standard
deviations, moments and moment generating
functions, linear regression and curve fitting,
limits theorems, stochastic processes, first and
second order characteristics, applications and
some advance topics.

7. Why this Course
• Probability
• Probability is a measure of the likeliness that an
event will occur.
This section deals with the chances of occurring any
particular phenomena. i.e. Electron emission, Telephone
calls, Radar detection, quality control, system failure, games
of chance, birth and death rates, Random walks, probability
of detection, Probability of false alarm, BER calculation,
optimal coding (Huffman) and many more.
• Moreover this study leads us to Random variables and
random processes.

8. Random Variables and Random Processes
• Mostly reader heard background hiss while
listening Radio.
• The waveform causing this hiss, when
measured on oscilloscope, would appear as a
randomly fluctuating voltage with time. Which is
called noise.
• In a television system this appears in the form
of snow.
• In SONAR randomly generated sea sounds
give rise to such noise.

9. Contd.
• The output voltage of a wind power generator
is a random due to random fluctuation of wind.
• Voltage from a solar detector are also random.
• Information of an antenna transmission.
All such phenomena must be described in
some probabilistic way .

10. Target
Hence, we are tempted to develop mathematical tools
for the analysis and quantitative characterization of
random signals. To be able to analyze random
signals, we need to understand random variables.
The resulting mathematical topics are: probability
theory, random variables and random (stochastic)
processes. In this course, we shall develop the
probabilistic characterization of random variables.

11. Set theory
• Revise at your own we have studied it
many times.

12. Chapter 1. Probability
Theory
1.1 Probabilities
1.2 Events

13. CHAPTER 1 Probability Theory
1.1 Probabilities
1.1.1 Introduction
• Statistics and Probability theory constitutes a branch of mathematics
for dealing with uncertainty
• Probability theory provides a basis for the science of statistical
assumption from data

14. CHAPTER 1 Probability Theory
1.1 Probabilities
1.1.2 Sample Spaces(1/3)
• Experiment : any process or procedure for which more than one
outcome is possible
• Sample Space
The sample space S of an experiment is a set consisting of all of the
possible experimental outcomes.

15. 1.1.2 Sample Spaces(2/3)
• Example 3: Software Errors
The number of separate errors in a particular piece of software can
be viewed as having a sample space
• Example 4: Power Plant Operation
A manager supervises the operation of three power plants, at any
given time, each of the three plants can be classified as either
generating electricity (1) or being idle (0).
{0 errors, 1errors, 2errors, 3errors,...}S =
{(0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)}S =

16. 1.1.2 Sample Spaces(3/3)
• GAMES OF CHANCE
- Games of chance commonly involve the toss of a coin, the roll of a
die, or the use of a pack of cards.
- The roll of a die
A usual six-sided die has a sample space
If two dice are rolled ( or, equivalently, if one die is rolled
twice), the sample space is shown in Figure 1.2.
{1,2,3,4,5,6}S =

17. 1.1.3 Probability Values(1/5)
• Probabilities
A set of probability values for an experiment with a sample space
consists of some probabilities
that satisfy
and
The probability of outcome occurring is said to be , and this is
written
( )i iP O p=
ipiO
1 2 1np p p+ + + =L
1 20 1,0 1, ,0 1np p p≤ ≤ ≤ ≤ ≤ ≤L
1 2, , , np p pL1 2{ , , , }nS O O O= L

18. 1.1.3 Probability Values(2/5)
• Example 3 Software Errors
Suppose that the numbers of errors in a software product have
probabilities
There are at most 5 errors since the probability values are zero for 6
or more errors.
The most likely number of errors is 2.
3 and 4 errors are equally likely in the software product.
(0 errors) 0.05, (1 error) 0.08, (2 errors) 0.35,
(3errors) 0.20, (4 errors) 0.20, (5 errors) 0.12,
( errors) 0, for 6
P P P
P P P
P i i
= = =
= = =
= ≥

19. 1.1.3 Probability Values(3/5)
• In some situations, notably games of chance, the experiments are
conducted in such a way that all of the possible outcomes can be
considered to be equally likely, so that they must be assigned
identical probability values.
• n outcomes in the sample space that are equally likely => each
probability value be 1/n.

20. 1.1.3 Probability Values(4/5)
• GAMES OF CHANCE
- A fair die will have each of the six outcomes equally likely.
- An example of a biased die would be one of which
In this case the die is most likely to score a 6, which will
happen roughly three times out of ten as a long-run average.
1
(1) (2) (3) (4) (5) (6)
6
P P P P P P= = = = = =

21. 1.1.3 Probability Values(5/5)
- If two die are thrown and each of the 36 outcomes is equally likely (
as will be the case two fair dice that are shaken properly), the
probability value of each outcome will necessarily be 1/36

22. 1.2 Events
1.2.1 Events and Complements(1/6)
• Events
An event A is a subset of the sample space S. It collects outcomes of
particular interest. The probability of an event is obtained
by summing the probabilities of the outcomes contained within the
event A.
• An event is said to occur if one of the outcomes contained within
the event occurs.
, ( ),A P A

23. 1.2.1 Events and Complements(2/6)
• A sample space consists of eight outcomes with a probability
value.
S
'
( ) 0.10 0.15 0.30 0.55
( ) 0.10 0.05 0.05 0.15 0.10 0.45
Notice that ( ) ( ) 1.
P A
P A
P A P A
= + + =
= + + + + =
′+ =

24. 1.2.1 Events and Complements(3/6)
• Complements of Events
The event , the complement of event A, is the event consisting
of everything in the sample space S that is not contained within the
event A. In all cases
• Events that consist of an individual outcome are sometimes referred
to as elementary events or simple events
A′
( ) ( ) 1P A P A′+ =

25. 1.2.1 Events and Complements(4/6)
• Example 3 Software Errors
Consider the event A that there are no more than two errors in a
software product.
A = { 0 errors, 1 error, 2 errors } S
and
P(A) = P(0 errors) + P(1 error) + P(2 errors)
= 0.05 + 0.08 + 0.35 = 0.48
P( ) = 1 – P(A) = 1 – 0.48 = 0.52
⊂
A′

26. 1.2.1 Events and Complements(5/6)
• GAMES OF CHANCE
- even = { an even score is recorded on the roll of a die }
= { 2,4,6 }
For a fair die,
- A = { the sum of the scores of two dice is equal to 6 }
= { (1,5), (2,4), (3,3), (4,2), (5,1) }
A sum of 6 will be obtained with
two fair dice roughly 5 times out of
36 on average, that is, on about
14% of the throws.
1 1 1 1
(even) (2) (4) (6)
6 6 6 2
P P P P= + + = + + =
1 1 1 1 1 5
( )
36 36 36 36 36 36
P A = + + + + =

27. 1.2.1 Events and Complements(6/6)
- B = { at least one of the two dice records a 6 }
11
( )
36
11 25
( ) 1
36 36
P B
P B
=
′ = − =