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Lecture 4 Discrete Random Variables: Definition and Probability Mass Function

Lecture 4 Discrete Random Variables: Definition and Probability Mass Function. Last Time Conditional Probability Independence Sequential Experiments &Tree Diagrams Counting Methods Independent Trials Reading Assignment : Sections 1.5-1.10. Probability & Stochastic Processes

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Lecture 4 Discrete Random Variables: Definition and Probability Mass Function

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  1. Lecture 4 Discrete Random Variables: Definition and Probability Mass Function Last Time Conditional Probability Independence Sequential Experiments &Tree Diagrams Counting Methods Independent Trials Reading Assignment: Sections 1.5-1.10 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008

  2. Lecture 4: DRV: definition and Mass Function Today • Reliability Problems • Definitions of Discrete Random Variables • Probability Mass Functions Tomorrow • Families of DRVs Reading Assignment: Sections 2.1-2.5 Homework #1 due 3/20 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 4- 2

  3. Lecture 4: Next Week • Discrete Random Variables • Cumulative Distribution Function (CDF) • Averages • Functions of DRV • Expected Value of a DRV • Variance and Standard Deviation Reading Assignment: Sections 2.6-2.9 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 4- 3

  4. What have you learned about conditional prob. and independence? • VLSI Testing example A semiconductor wafer has M VLSI chips on it and these chips have the same circuitry. Each VLSI chip consists of N interconnected transistors. A transistor may fail (not function properly) with a probability p because of its fabrication process, which we assume to be independent among individual transistors. A chip is considered a failure if there are n or more transistor failures. Derive the probability that a chip is good. Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 4- 4

  5. What have you learned about conditional prob. and independence? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 • Root cause diagnosis Now suppose that the value of a current I of the chip depends on transistor 1. If transistor1 fails, we will observe an abnormal I value with a probability q and a normal I value with a probability 1-q; if transistor1 is good, we will observe an normal I value with a probability r and an abnormal I value with a probability 1-r. What is the probability that you measure an abnormal I value? When the I value you measured is normal, what is the probability that transistor 1 actually fails? 4- 5

  6. What have you learned? • Ans: b/(a+b), why? • Solution: • Let pi be the probability that a gambler starts with i dollars and run out of money eventually. • After a coin toss, the gambler either wins or loses and starts from i+1 or i-1 dollars, so • p(i )= (1/2) p(i+1) + (1/2) p(i-1) . •  Also, p0 = 1, pa+b = 0, so • p(i+1 )- p(i ) = p(i)– p(i-1) •  Finally, we have • p(i()= (a+b-i)/(a+b), • pa = b/(a+b). •  For an unfair coin toss game, we can also find the probability that gambler A runs out of money Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 • Gambler’s Ruin Problem Two gamblers A and B with a and b dollars play the game of fair coin toss with wager of $1 per game until one of them runs out of money. What is the probability that A runs out of money? 4- 6

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  14. Example 2.A1 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 (1) Toss a coin (2) Gender at birth (3) Random walk Q: Probability space of each experiment? 4- 15

  15. Example 2.A1 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 (4) Throw a dart and the distance to the center of the disk with radius R? (5) A particle falls on a semiconductor wafer with radius R? Q: Probability space of each experiment? 4- 16

  16. Example 2.A2 Q: How are X and Y related to events? Q:How would you associate probability to X and Y? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 Toss a die Q: Could you define two random variables X and Y? 4- 20

  17. Example 2.A3 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 Draw three cards from a deck of 52 cards with replacement. Let X be the number of spades in these three cards. What is the PMF of X? 4- 25

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