ECE313.Lecture23

ECE313.Lecture23 - ECE 313 Probability with Engineering Applications Continuous Random Variables II Professor Dilip V Sarwate Department of

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Continuous Random Variables II 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 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Continuous random variables Definition: A continuous random variable X is one whose CDF F X (u) is continuous at all u, < u < differentiable at all u (except possibly at a set of points u 1 < u 2 < … u n < ) More precisely, any finite-length interval contains at most finitely many points where F X (u) is not differentiable
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ECE 313 - Lecture 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 The derivative of the CDF The derivative of the CDF of a continuous random variable X exists for almost all real numbers u Definition: The probability density function (pdf) of a continuous random variable X is F X (u), if F X (u) is differentiable f X (u) = any number ≥ 0 if CDF is non-diff. d du
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ECE 313 - Lecture 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 What values can X take on? For every real number u, P{ X = u} = 0 The set of all possible values that X can take on is indicated by defining f X (u) explicitly (by means of some formula, say) at those values The set of all possible values that X cannot take on is indicated by the word elsewhere in the definition of the pdf, as in f X (u) = 0 elsewhere
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ECE 313 - Lecture 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 The pdf versus the pmf A discrete random variable defines a set of point masses on the axis: total mass = 1 In contrast, a continuous random variable defines a spread of the total probability mass of 1 along the axis There is no mass at any point The pdf of a continuous random variable tells the density of the mass at each point
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ECE 313 - Lecture 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 So, what’s a pdf mean, anyway? The probability density function (pdf) of a continuous random variable tells us the density of the probability mass at each point on the axis pdf is measured in units of mass/length The concept of the pdf is analogous to the more common concepts of the mass density, charge density, etc
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ECE 313 - Lecture 23 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 pdfs are not what you think they are The probability density function (pdf) of a continuous random variable is not , by itself, a probability Example: f X (u) = 3u 2 for 0 ≤ u ≤ 1, and 0 elsewhere This pdf has value 3 at u = 1 and 3/4 at 0.5 This means the probability mass is four
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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.Lecture23 - ECE 313 Probability with Engineering Applications Continuous Random Variables II Professor Dilip V Sarwate Department of

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