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ECE313.Lecture30

# ECE313.Lecture30 - ECE 313 Probability with Engineering...

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Joint Distributions of Random Variables Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 33 Random variables — a review A random variable assigns a real number to each outcome in the sample space The random variable X is said to map the outcome ϖ ∈ Ω to the real number X ( ϖ ) The function X from to is fixed; ϖ is always mapped to the same number X ( ϖ ) Randomness arises from not knowing which outcome will occur on a trial and hence not knowing what numerical value of X will be observed on that trial
ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 33 Probabilistic description The probabilistic behavior of any random variable X can be described via its CDF F X (u) = P{ X ≤ u} CDFs are cumbersome to use, but are an important concept in problem-solving Discrete random variables are usually discussed in terms of their pmfs Continuous random variables are usually discussed in terms of their pdfs

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ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 33 Multiple random variables We want to study the joint probabilistic behavior of many random variables defined on the same sample space Different random variables correspond to different physical parameters; their joint behavior is often of great interest Some experimental observations result in vectors or sequences of numbers — thus it is important to have a theory for dealing with random vectors
ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 33 The joint behavior is important! Example: The time of arrival of requests for files at a server is a Poisson process with given arrival rate. The sizes of the files requested are random Large files are more likely to be requested at some particular times of the day than at other times What is the distribution of the average output traffic per second? What is P{buffer overﬂow}?

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ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 33 How many RVs did you say? The generalization from one random variable to two random variables is the most challenging intellectual concept Once the two random variable case is understood, the extension of the ideas to many random variables is easy We discuss the two-random-variable case in detail and indicate brieﬂy how the theory extends to more than two variables
ECE 313 - Lecture 30 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 33 The random point in the plane I Let X and Y denote two random variables

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