ECE313.Lecture13

# ECE313.Lecture13 - ECE 313 Probability with Engineering...

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The Theorem of Total Probability 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 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 39 Introduction The conditional probability of an event B given that event A occurred is our revised estimate of the chances that B occurred in light of partial knowledge of the outcome of the experiment, viz. knowing that A occurred To avoid trivialities, we assume that A, sometimes called the conditioning event, has nonzero probability
ECE 313 - Lecture 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 39 Defnition oF conditional probability The conditional probability of B given A is denoted by P(B | A) Read this as “the probability of B given A” or “the probability of B conditioned on A” DeFnition: If P(A) > 0, P(B | A) is deFned as P(B | A) = P(AB) P(A) P(B | A) can be larger than, smaller than, or the same as P(B)

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ECE 313 - Lecture 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 39 Consistent with various models The defnition oF conditional probability is consistent with classical approach to probability relative Frequency approach Conditional probabilities can also be discussed For events defned in terms oF random variables P{ X = k | X > n}? or P{ X ≤ k | a < X < b}?
ECE 313 - Lecture 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 39 Geometric RVs are memoryless Let X denote a geometric random variable with parameter p For k > 0, P{ X = k+r | X > r} = P{ X = k} Given that the event { X > r} has occurred, that is, the frst r trials ended in a “±ailure”, the probability that we need to wait ±or an additional k trials to observe the frst success is the same as P{ X = k} It’s as i± the frst r trials are ±orgotten!

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ECE 313 - Lecture 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 39 Binomial random variables Let X denote a binomial random variable with parameters (n, p) GIven the event { X = k} has occurred, the conditional probability that the j-th trial resulted in a success is k/n, independent of the value of p The conditional probability of successes on the i-th and j-th trials is k(k–1)/[n(n–1)] and so on
ECE 313 - Lecture 13 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 39 Axioms are satisfed Conditional probabilities are a probability measure, that is, they satisfy the axioms of probability theory All the consequences of the axioms (rules of probability) also apply to conditional probabilities Caveat: Everything must be conditioned on the same event. No mixing and matching allowed

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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.Lecture13 - ECE 313 Probability with Engineering...

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