1. Lecture 6 Discrete Random Variables: Definition and Probability Mass Function Last Time Families of DRVs Cumulative Distribution Function (CDF) Averages Functions of RDV Reading Assignment: Sections 2.1-2.7 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 6 - 1

2. Lecture 6: DRV: CDF, Functions, Exp. Values Today Discrete Random Variables Functions of DRV (cont.) Expectation of DRV Variance and Standard Deviation Conditional Probability Mass Function Continuous Random Variables (CRVs) CDF Tomorrow Probability Density Functions (PDF) Expected Values Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 6 - 2

3. Lecture 6: 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 6- 3

4. What have you learned? Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_03_2008 Clarifications Axiom 3 Pascal PMF Pascal RV.doc Limit to exponential Limit to Exponential.doc 6- 4

5. St. Petersburg Paradox 16 - 5 A paradox presented to the St Petersburg Academy in 1738 by the Swiss mathematician and physicist Daniel Bernoulli (1700–82).paradox A coin tossed if it falls heads then the player is paid one rouble and the game ends. If it falls tails then it is tossed again, and this time if it falls heads the player is paid two roubles and the game ends. This process continues, with the payoff doubling each time, until heads comes up and the player wins something, and then it ends.

6. St. Petersburg Paradox: Discussion 16 - 6 Q: How much should a player be willing to pay for the opportunity to play this game? the game's expected value = (½)(1) + (¼)(2) + (⅛)(4) +… = ? Q: Do you want to play? Probably not!  a high probability of losing everything For example, 50 %chance of losing it on the very first toss  the principle of maximizing expected value  Bernoulli's introduction of ‘moral worth’(and later utility (1).)utility (1)

26. Tank Number Estimation 16 - 26 Tank Example Tank Example.doc