ECE313.Lecture29

# ECE313.Lecture29 - ECE 313 Probability with Engineering...

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Decision-making under uncertainty IV 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 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 Conditional CDF and pdf of X Given that event A (with P(A) > 0) has occurred, the conditional CDF of X is F X | A (u | A) = P{ X ≤ u | A} P{ X ≤ u | A} = P({ X ≤ u} A)/P(A) is the conditional probability that { X ≤ u} given that the event A occurred If X is a continuous random variable, then f X | A (u | A), the conditional pdf of X is the derivative of the conditional CDF F X | A (u | A)
ECE 313 - Lecture 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 Just as good as the regular kind … Conditional CDFs have all the properties of ordinary CDFs — they are monotone nondecreasing, right-continuous functions with limits 0 at – and 1 at Conditional pdfs have all the properties of ordinary pdfs — they are non-negative functions of total area 1, and P{u ≤ X ≤ u+ u | A} = P{u ≤ X u+ u}/P(A) ≈ f X | A (u | A)• u

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ECE 313 - Lecture 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 When A is specified in terms of X… Sometimes event A is specified in terms of the random variable X , e.g. A = {c < X < d} P(A) = area under the unconditional pdf f X (u) between c and d f X | A (u | A) = f X (u)/P(A) if u A f X | A (u | A) = 0 if u A c P(A) = area under f X (u) in region A is a “normalizing factor” that makes f (u)/P(A) = f (u | A) a valid density
ECE 313 - Lecture 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 Conditional mean and variance of X We can calculate means and variances using conditional pdfs The mean of the conditional pdf is called the conditional mean E[ X | A] The variance of the conditional pdf is called the conditional variance var( X | A) Instead of using the unconditional pdf f X (u), use the conditional pdf f X | A (u | A) in the various integrals

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ECE 313 - Lecture 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 Unconditional pdf from conditional Let {A 1 , A 2 , … , A n , … } be a (countable) partition of The theorem of total probability gives the unconditional distributions in terms of the conditional distributions (cf. Lecture 14) F X (u) = F X | A i (u | A i )•P(A i ) f X (u) = f X | A i (u | A i )•P(A i ) E[ X ] = E[ X | A i ]•P(A i )
ECE 313 - Lecture 29 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 36 Unconditional variance of X

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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.Lecture29 - ECE 313 Probability with Engineering...

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