Mathematics for Computer Science
18.4. Great Expectations 753 18.4.4 Alternate Definition of Expectation There is another standard way to define expectation. The ...
Chapter 18 Random Variables754 Definition 18.4.4.Theconditional expectationExŒRjAçof a random variableR given eventAis: ExŒRjAçW ...
18.4. Great Expectations 755 Proof. ExŒRçD X r 2 range.R/ rPrŒRDrç (by 18.3) D X r r X i Pr RDrjAi PrŒAiç (Law of Total Pr ...
Chapter 18 Random Variables756 Plugging (18.6) and (18.7) into (18.5): ExŒCçD 1 pC.1CExŒCç/.1p/ DpC 1 pC.1p/ExŒCç D 1 C.1p/ExŒC ...
18.4. Great Expectations 757 For the record, we’ll state a formal version of this result. A random variable likeCthat counts ste ...
Chapter 18 Random Variables758 you guess right? Eric guesses right? no yes no yes 1=2 1=2 1=2 1=2 1=2 1=2 yes 1=2 no 1=2 yes 1=2 ...
18.4. Great Expectations 759 tree diagram to compute your expected return. The tree diagram is shown in Fig- ure 18.6. The “payo ...
Chapter 18 Random Variables760 you guess right? Eric guesses right? no yes no yes 1=2 1=2 1=2 1=2 1=2 1=2 yes 0 no 1 yes 1 no 0 ...
18.4. Great Expectations 761 The payoffs for each outcome are the same in Figures 18.6 and 18.7, but the probabilities of the ou ...
Chapter 18 Random Variables762 was selected by a large fraction of the population. Apparently many people think the same way; th ...
18.5. Linearity of Expectation 763 Theorem 18.5.2.For random variablesR 1 ,R 2 and constantsa 1 ;a 22 R, ExŒa 1 R 1 Ca 2 R 2 çDa ...
Chapter 18 Random Variables764 his own hat back is1=n. There are many different probability distributions of hat permutations wi ...
18.5. Linearity of Expectation 765 Proof. DefineRito be the indicator random variable forAi, whereRi.!/D 1 if w 2 AiandRi.!/D 0 ...
Chapter 18 Random Variables766 By Theorem 18.5.4, ExŒJçD Xn iD 1 PrŒJiçDpn: (18.12) That really was easy. If we flipnmutually in ...
18.5. Linearity of Expectation 767 Let’s partition the sequence into 5 segments: blue „ƒ‚... X 0 green „ƒ‚... X 1 green red „ ƒ‚ ...
Chapter 18 Random Variables768 We can use equation (18.14) to answer some concrete questions. For example, the expected number o ...
18.5. Linearity of Expectation 769 18.5.6 A Gambling Paradox One of the simplest casino bets is on “red” or “black” at the roule ...
Chapter 18 Random Variables770 Now the dollar amount you win in any gambling session is X^1 nD 1 Bn; and your expected win is Ex ...
18.5. Linearity of Expectation 771 18.5.8 Expectations of Products While the expectation of a sum is the sum of the expectations ...
Chapter 18 Random Variables772 Hint:LetA^1 WWDAandA^0 WWDA, so the eventŒIADcçis the same asAcfor c2f0;1g; likewise forB^1 ;B^0. ...
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