Lecture 6

Discrete Random Variables: Definition

and Probability Mass Function

Last Time Families of DRVs Cumulative Distribution Function (CDF) Averages Functions of RDVReading Assignment: Sections 2.1-2.7

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_03_20086 - 1

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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_03_20086 - 2

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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_03_20086- 3

What have you learned?

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_03_2008

Clarifications

Axiom 3

Pascal PMF

Pascal RV.doc

Limit to exponential

Limit to Exponential.doc

6- 4

St. Petersburg Paradox

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A paradox presented to the St Petersburg Academy in 1738 by the Swiss mathematician and physicist Daniel Bernoulli (1700–82).

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.

St. Petersburg Paradox: Discussion

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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 everythingFor example, 50 %chance of losing it on the very first tossthe principle of maximizing expected valueBernoulli's introduction of ‘moral worth’(and later utility (1).)

Tank Number Estimation

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Tank ExampleTank Example.doc