1. Lecture 5 Discrete Random Variables: Definition and Probability Mass Function Last Time Reliability Problems Definitions of Discrete Random Variables Probability Mass Functions Families of DRVs Reading Assignment: Sections 2.1-2.3 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 5- 1

2. Lecture 5: DRV: CDF, Functions, Exp. Values Today Families of DRVs (Cont.) Cumulative Distribution Function (CDF) Averages Functions of DRV Tomorrow Functions of DRV (Cont.) Expected Value of a DRV Reading Assignment: Sections 2.4-2.7 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 5- 2

3. Lecture 5: Next Week Discrete Random Variables Variance and Standard Deviation Conditional Probability Mass Function Continuous Random Variables (CRVs) CDF Probability Density Functions (PDF) Expected Values Families of CRVs Reading Assignment: Sections 2.8-3.4 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 5- 3

4. What have you learned about DRV? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 5- 4 Example: Taiwan’s Presidential Election of 3/22 D.R.V. Poll Gambling

5. What have you learned? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 5- 5 Example Alex, Ben, and Tim are three prisoners, one of whom is scheduled to die. If a jailer told Alex that which one of Ben and Tim is going to be freed, will this change the probability of Alex dying? Let A, B, T, and J be the event that Alex, Ben, and Tim will die and the event that a jailer told Alex that Tim will be freed. Then P(A|J) = P(J|A)P(A)/[P(J|A)P(A) + P(J|B)P(B) + P(J|T)P(T)] = (1/2)(1/3)/[(1/2)(1/3) + 1(1/3) + 0(1/3)] = 1/3. Q: Relation to independence? Poll Gambling

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8. Example 2.A2 Toss a die Q: Could you define two random variables X and Y? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 4- 8 Q: How are X and Y related to events? Q:How would you associate probability to X and Y?

51. Example: Eggs in Different Baskets You have $100,000 to invest Investment opportunity 0.2 prob. you get 10 times of your investment 0.8 prob. you lose the your investment Q1: Let X be the D.R.V. of the $ you earn. E[X] = ? Do you want to invest on this opportunity ? Q2: Now you have 100,000,000. Would you invest? Q3: If there are 100 same kind of opportunities, what would you do? E[X] = ?