ECE313.Lecture22

# ECE313.Lecture22 - ECE 313 Probability with Engineering...

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Continuous Random Variables I 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 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 44 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 –
ECE 313 - Lecture 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 44 CDF of a discrete random variable The CDF of any discrete random variable is a staircase function If X takes on values u 1 , u 2 , … u n , with probabilities p(u 1 ), p(u 2 ), … , p(u n ), then the CDF has jumps whose heights are p(u 1 ), p(u 2 ), … , p(u n ), at locations u 1 , u 2 , … u n , respectively

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ECE 313 - Lecture 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 44 CDF of a discrete random variable The CDF of any discrete random variable is a staircase function We have defined the u i to be an increasing sequence u 1 < u 2 < … < u n < … For all u in the range u i ≤ u < u i+1 , F X (u) has value p(u j ) j = 1 i F X (u) jumps in value by p(u i+1 ) at u = u i+1
ECE 313 - Lecture 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 44 Negative thoughts about the CDF The CDF of a discrete random variable X is far more cumbersome than the pmf The CDF contains no information that we could not have deduced from the pmf So, why bother with the CDF at all? The concept of the CDF of X is a very general one that applies to every random variable, discrete or continuous or mixed…

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ECE 313 - Lecture 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 44 The power of positive thinking The concept of the CDF of X is a very general one that applies to every random variable, discrete or continuous or mixed… Every random variable X has a CDF that is defined in exactly the same way: F X (u) = P{ X ≤ u}, < u < But, discrete random variables have pmfs,
ECE 313 - Lecture 22 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 44 More power of positive thinking Common properties of random variables, i.e. those shared by all random variables, can be stated in terms of the CDF instead of having a different formula for each type of random variable The CDF is a unifying concept for all kinds of random variables

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## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture22 - ECE 313 Probability with Engineering...

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