Mathematics for Computer Science
17 Random Variables Thus far, we have focused on probabilities of events. For example, we computed the probability that you win ...
Chapter 17 Random Variables574 Similarly,Mis a function mapping each outcome another way: M.HHH/ D 1 M.THH/ D 0 M.HHT/ D 0 M.THT ...
17.2. Independence 575 LikewiseŒMD1çis the eventfT T TCHHHgand has probability1=4. More generally, any assertion about the value ...
Chapter 17 Random Variables576 ThenH 1 is independent ofM, since PrŒMD1çD1=4DPr MD 1 jH 1 D 1 DPr MD 1 jH 1 D 0 PrŒMD0çD ...
17.3. Distribution Functions 577 3= 36 6= 36 x 2 V 2 3 4 5 6 7 8 9 10 11 12 PDFT.x/ Figure 17.1 The probability density function ...
Chapter 17 Random Variables578 0 1=2 1 x 2 V 0 1 2 3 4 5 6 7 8 9 10 11 12 : : : CDFT.x/ Figure 17.2 The cumulative distribution ...
17.3. Distribution Functions 579 17.3.2 Uniform Distributions A random variable that takes on each possible value in its codomai ...
Chapter 17 Random Variables580 Intuition Behind the Winning Strategy Amazingly, there is a strategy that wins more than 50% of t ...
17.3. Distribution Functions 581 Step 1: Find the sample space. You either choosextoo low (< L), too high (> H), or just r ...
Chapter 17 Random Variables582 Step 4: Compute event probabilities. The probability of the event that you win is the sum of the ...
17.3. Distribution Functions 583 f 20 .k/ 0:18 0:16 0:14 0:12 0:10 0:08 0:06 0:04 0:02 0 k 0 5 10 15 20 Figure 17.4 The pdf for ...
Chapter 17 Random Variables584 f20;:75.k/ 0:25 0:2 0:15 0:1 0:05 0 k 0 5 10 15 20 Figure 17.5 The pdf for the general binomial d ...
17.4. Great Expectations 585 17.4 Great Expectations Theexpectationorexpected valueof a random variable is a single number that ...
Chapter 17 Random Variables586 Now, ExŒSçDEx 1 R D 1 1 1 6 C 1 2 1 6 C 1 3 1 6 C 1 4 1 6 C 1 5 1 6 C 1 6 1 6 D 4 ...
17.4. Great Expectations 587 Proof. SupposeRis defined on a sample spaceS. Then, ExŒRçD X ! 2 S R.!/PrŒ!ç (Def 17.4.1 of expecta ...
Chapter 17 Random Variables588 17.4.5 Conditional Expectation Just like event probabilities, expectations can be conditioned on ...
17.4. Great Expectations 589 Proof. ExŒRçD X r 2 range.R/ rPrŒRDrç (by 17.2) D X r r X i Pr RDrjAi PrŒAiç (Law of Total Pr ...
Chapter 17 Random Variables590 Plugging (17.5) and (17.6) into (17.4): ExŒCçD 1 pC.1CExŒCç/.1p/ DpC 1 pC.1p/ExŒCç D 1 C.1p/ExŒC ...
17.4. Great Expectations 591 of the firsti 1 hours, times the probability,p, that it does crash in theith hour. So ExŒCçD X i 2 ...
Chapter 17 Random Variables592 For example, if you are flipping a fair coin and you win $1 for heads and you lose $1 for tails, ...
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