ECE313.Lecture24

ECE313.Lecture24 - ECE 313 Probability with Engineering...

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Continuous Random Variables III 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 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 The pdf of a continuous RV The pdf f X (u) of a continuous random variable is the derivative of its CDF The pdf has two properties f X (u) ≥ 0 Total area under the pdf curve f X (u) from to is 1 Fancy statement: f X (u) du = 1
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Other properties enjoyed by the pdf The pdf is a nonnegative function that has unit area between the pdf curve and the horizontal axis f X (+ ) = lim u + f X (u) = 0 f X (– ) = lim u f X (u) = 0 Compare to F X (+ ) = 1 and F X (– ) = 0
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Probabilities from the pdf P{a ≤ X ≤ a + δ a} ≈ f X (a)• δ a P{a < X < b} = Area under the pdf curve between a and b a b = f X (u) du P{ X = u} = 0 for all real numbers u P{a < X < b}, P{a < X ≤ b}, P{a ≤ X < b}, and P{a ≤ X ≤ b} all have the same value
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 Relationship between CDF and pdf The value of the CDF at the point u = 5 is F X (5) = area under pdf f X (u) from – to 5 The CDF is not the antiderivative (or indefinite integral) of the pdf; it is the definite integral Re-read your calculus books to refresh your understanding of the Fundamental Theorem of Calculus
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 ECE 313 Survival Guide Always, always, always sketch the pdf (or CDF) curve before you do anything else Indicate the desired probability as an area (e.g. by shading) on the sketch If you use an integral to find the area, set the limits with the help of the sketch Do not use indefinite integrals All integrals must have limits
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 Expectation of an arbitrary RV u 1 E[ X ] = [1 – F X (u)] du F X (u) du 0 0 The expected value E[ X ] of an arbitrary random variable X can be defined as F X (u) = blue area – orange area
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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 40 Expectation of a continuous RV E[ X ] = [1 – F X (u)] du F X (u) du 0 0 E[ X ] = u•f X (u) du
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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.Lecture24 - ECE 313 Probability with Engineering...

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