# Chapter1 - EEM3066 Random Processes Queueing Theory Chapter...

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EEM3066: Random Processes & Queueing Theory Chapter 1: Probability & RV Page 1 of 32 Chapter 1: Probability & Random Variable Concepts Objective: To review the theory and concept of probability and random variable At the completion of the chapter, students should be able to: Understand the concepts of random variables and transformation. Contents 1.1 Probability Theory 1.1.1 Introduction 1.1.2 Set Theory, Sample Space and Events 1.1.3 Concepts of Probability 1.1.4 Conditional Probability 1.1.5 Total Probability 1.1.6 Bayes’ Theorem 1.2 Random Variable 1.2.1 Introduction 1.2.2 Random Variable 1.2.3 Distribution Function/ Cumulative Distribution Function 1.2.4 Probability Density Function (PDF) 1.2.5 Commonly used Distribution and Density Functions 1.2.6 Change of Random Variable 1.2.7 Expectation of RV 1.2.8 Multiple RVs 1.2.9 Marginal Distribution and Density Functions 1.2.10 Independent Random Variable 1.2.11 Joint Moments 1.2.12 T ransformation of Two Random Variables 1.3 Summary References

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EEM3066: Random Processes & Queueing Theory Chapter 1: Probability & RV Page 2 of 32 1.1 Probability Theory 1.1.1 Introduction Probability theory deals with the averages of mass phenomena whether occurring simultaneously or sequentially. 1.1.2. Set Theory, Sample Space and Events Set: A set is a collection of well-defined objects called elements of the set. For example, the set of outcomes of tossing a coin is S ={Head, Tail}. Union of Sets : Let X and Y be two sets .Then the Union of Sets X and Y is denoted as = Y X {Elements of both the sets X and Y } The set of all possible outcomes of a random experiment is called the sample space S . Any subset of the sample space is called an event . Intersection of Sets : The intersection of Sets X and Y is denoted as = Y X {Elements common to both the sets X and Y } In the Figure 1.1, (stars) represent all the outcomes. The dotted closed curve encompassing some outcomes represents an event and rectangle encompassing all possible outcomes is the Sample space Events Outcome Figure 1.1 : Venn diagram representation of Sample Space, Events and Outcomes S S Event 1 Event 2 Figure 1.2 : Mutually exclusive events
EEM3066: Random Processes & Queueing Theory Chapter 1: Probability & RV Page 3 of 32 1.1.3 Concepts of Probability There are different ways of defining Probability. a) Classical approach : we do not perform any experiments, but define the probability of an event as the ratio of the number of outcomes of the event to the total number of outcomes in the sample space. If N X is the number of outcomes of an event X and N is the total number of outcomes in sample space S, then probability of the event X occurring is ( 29 N N X P X = (1.1) Consider the rolling of a dice and observing the number on top face. The sample space will be S = {1, 2, 3, 4, 5, 6} or N =6 Consider an event namely the number on top face is divisible by 2.

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