ECE313.Lecture21

# ECE313.Lecture21 - ECE 313 Probability with Engineering...

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The Cumulative Distribution Function 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 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 34 Discrete Random Variables Recall the generic description of a discrete random variable X X takes on values u 1 , u 2 , … u n , with probabilities p(u 1 ), p(u 2 ), … , p(u n ), The number of different values is either finite (the list of values terminates at u n for some n > 1) or is countably infinite The values are discretely spaced : u 1 < u 2 < … < u n < …
ECE 313 - Lecture 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 34 Cumulative Distribution Function The cumulative (probability) distribution function or CDF of a random variable X is denoted by F(u) or F X (u) Definition: F X (u) = P{ X ≤ u}, < u < The CDF is a real-valued function defined for all real number values of its argument u Cumulative because the value of the CDF at u is the total probability mass from –

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ECE 313 - Lecture 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 34 Pejorative remarks on notation Most textbook authors (including Ross) use x as the argument of the CDF of a random variable X They write P{ X ≤ x} instead of P{ X ≤ u} and F X (x) instead of F X (u) This notation causes many problems in students’ minds What’s the difference between X and x? Which is the RV and which the number?
ECE 313 - Lecture 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 34 Doctors and engineers alike… The notation P{ X ≤ x} instead of P{ X ≤ u} and F X (x) instead of F X (u) causes many problems in students’ handwriting As the semester wears on (or even right away for the calligraphically challenged), the distinction between X and x is lost, as X decreases and x increases in size All your formulas will look like P{x ≤ x} by the end of the semester and you won’t know what it means!

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ECE 313 - Lecture 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 34 Say it loud and clear… The notation P{ X ≤ x} instead of P{ X ≤ u} and F X (x) instead of F X (u) causes many problems in ordinary speech The notation is hard to read aloud P{ X ≤ x} is “probability that big ecks is less than or equal to small ecks”? F X (x) is read as “eff sub ecks of ecks”? or should it be read as “eff sub big ecks of little ecks”?
ECE 313 - Lecture 21 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 34 Keeping things straight… In these days of global-search-and- replace it is not hard to change all occurrences of (x) to (u) and x} to u} etc In these slides, random variables are

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ECE313.Lecture21 - ECE 313 Probability with Engineering...

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