Random Variable and Stochastic Processes

Instructor: Muhammad Shahid Iqbal

(m.shahid@uog.edu.pk)

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.

Course DetailsCourse Code EE-Credit Hours 3(3+0)Title Random Variables and Stochastic processes

Instructor Muhammad Shahid IqbalOffice hours: Tuesday 3:00-4:30, Thursday 12:00-1:30Office and Consulting Timings Room Number 207 EE DepartmentGrading PolicySessional 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

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.

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.

CourseSet 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.

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.

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.

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 .

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.

Set theory

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

Chapter 1. Probability Theory

1.1 Probabilities

1.2 Events

CHAPTER 1 Probability Theory1.1 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

CHAPTER 1 Probability Theory1.1 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.

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 =

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 =

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=

ipiO1 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

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

= = == = == ≥

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.

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)

6P P P P P P= = = = = =

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

1.2 Events1.2.1 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

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

= + + == + + + + =

′+ =

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′+ =

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′

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 2P P P P= + + = + + =

1 1 1 1 1 5( )

36 36 36 36 36 36P A = + + + + =

1.2.1 Events and Complements(6/6)

- B = { at least one of the two dice records a 6 }

11( )

3611 25

( ) 136 36

P B

P B

=

′ = − =