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ELEC 303 – Random Signals

ELEC 303 – Random Signals. Lecture 2 – Conditional probability Farinaz Koushanfar ECE Dept., Rice University Aug 27, 2009. Lecture outline. Reading: Sections 1.3, 1.4 Review Conditional probability Multiplication rule Total probability theorem Bayes rule.

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ELEC 303 – Random Signals

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  1. ELEC 303 – Random Signals Lecture 2 – Conditional probability FarinazKoushanfar ECE Dept., Rice University Aug 27, 2009

  2. Lecture outline • Reading: Sections 1.3, 1.4 • Review • Conditional probability • Multiplication rule • Total probability theorem • Bayes rule

  3. Probability theory -- review • Mathematically characterizes random events • Defined a sample space of possible outcomes • Probability axioms: • (Nonnegativity) 0≤P(A)≤1 for every event A • (Additivity) If A and B are two disjoint events, then the probability P(AB)=P(A)+P(B) • (Normalization) The probability of the entire sample space  is equal to 1, i.e., P()=1  A B

  4. Discrete/continuous models -- review • Discrete: finite number of possible outcomes • Enumerate the possible scenarios and count • Continuous: the sample space is continuous • The probability of a point event is zero • Probability=area in the sample space

  5. Conditional probability example 1 Student stage: (FR), (SO), (JU), (SI) Student standing: probation (P), acceptance (A), honor (H) Example courtesy of www.ms.uky.edu/~lee/amsputma507

  6. Example (cont’d) What is the probability of a JU student? What is the probability of honors standing if the student is a FR? What is the probability of probation for a SU? P(A|B) is probability of A given B B is the new universe

  7. Conditional probability  A B Definition: Assuming P(B)0, then P(A|B) = P(AB) / P(B) Consequences: If P(A)0 and P(B)0, then P(AB) = P(B).P(A|B) = P(A).P(B|A)

  8. Conditional probability example 2 • What is the probability of both dices showing odd numbers given that their sum is 6? {1,5},{5,1},{2,4},{4,2},{3,3} {1,5},{5,1},{3,3} • Let B is the event: min (X,Y)=3, Let M = max{X,Y}. What are the probabilities for M over all of its possible values?

  9. Conditional probability example 3 Radar detection vs. airplane presence What is the probability of having an airplane? What is the probability of airplane being there if the radar reads low? When should we decide there is an airplane and when should be decide there is none? Slide courtesy of Prof. Dahleh, MIT

  10. Sequential description P(L|A)=0.066 Multiplication rule: Assuming that all of the conditioning events have positive probability, P(M|A)=0.26 P(A)=0.3 P(H|A)=0.66 (A): aircraft present, (Ac): aircraft absent (L): low, (M): medium, (H): high P(L|Ac)=0.643 P(M|Ac)=0.286 P(Ac)=0.7 P(H|Ac)=0.071

  11. Conditional probability example 4 Three cards are selected from a deck of 52 cards without replacement. Find the probability that none of the drawn cards is a picture, i.e., (J,Q,K)  drawn set

  12. The Monty hall problem • You are told that a prize is equally likely behind any of the 3 closed doors • You randomly point to one of the doors • A friend opens one of the remaining 2 doors, after ensuring that the prize is not behind it • Consider the following strategies: • Stick to your initial choice • Switch to the other unopened door Picture courtesy of http://hight3ch.com

  13. Total probability theorem B A1B Bc A1 B A2B Divide and conquer Partition the sample space into A1, A2, A3 For any even B: P(B) = P(A1B) + P(A2B) + P(A3B) = P(A1)P(B|A1)+P(A2)P(B|A2)+P(A3)P(B|A3) A2 Bc A3 A3B B Figure courtesy of Bertsekas&Tsitsiklis, Introduction to Probability, 2008 Bc

  14. Radar example 3 (cont’d) P(Present) = 0.3 P(Medium|Present)=0.08/0.3 P(Present|low)=0.02/0.47 Example courtesy of Prof. Dahleh, MIT

  15. Radar example 3 (cont’d) Given the radar reading, what is the best decision about the plane? Criterion for decision: minimize “probability of error” Decide absent or present for each reading Example courtesy of Prof. Dahleh, MIT

  16. Radar example 3 (cont’d) Error={Present and decision is absent} or {Absent and decision is present} Disjoint events P(error)=0.02+0.08+0.05 Example courtesy of Prof. Dahleh, MIT

  17. Extended radar example • Threat alert affects the outcome P(…|Threat) P(…|No threat) P(Threat)=Prior probability of threat = p Example courtesy of Prof. Dahleh, MIT

  18. Extended radar example A=Airplane, R=Radar reading If we let p=P(Threat), then we get: Example courtesy of Prof. Dahleh, MIT

  19. Extended radar example Given the radar registered high and a plane was absent, what is the probability that there was a threat? How does the decision region behave as a function of p? Example courtesy of Prof. Dahleh, MIT

  20. Extended radar example Example courtesy of Prof. Dahleh, MIT

  21. Bayes rule Let A1,A2,…,An be disjoint events that form a partition of the sample space, and assume that P(Ai) > 0, for all i. Then, for any event B such that P(B)>0, we have The total probability theorem is used in conjunction with the Bayes rule:

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