l13_expectvar

# l13_expectvar - Expectation & Variance Reading: Ross, Ch...

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03/22/2009 CS206 - Intro. to Discrete Structures II 1 Reading: Ross, Ch 4., Sec. 3-5. Monday, March 23, 2009

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03/22/2009 CS206 - Intro. to Discrete Structures II 2 3 tosses of a coin Recall the problem of tossing a coin three times and the profit the player earns from this game: Outcome P(O) Profit HHH 1/8 3 HHT 1/8 1 HTH 1/8 1 HTT 1/8 -1 THH 1/8 1 THT 1/8 -1 TTH 1/8 -1 TTT 1/8 -3 Profit X P(X=x) 3 1/8 1 3/8 -1 3/8 -3 1/8 Monday, March 23, 2009
03/22/2009 CS206 - Intro. to Discrete Structures II 3 Expected profit • What is the average or expected profit in this game? Average profit = ( 3 + 1 + 1 + -1 + 1 + -1 + -1 + -3 ) / 8 = 0 Another way of computing this average profit: Average profit = 3 ¢ 1/8 + 1 ¢ 3/8 + -1 ¢ 3/8 + -3 ¢ 1/8 = 0 Monday, March 23, 2009

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03/22/2009 CS206 - Intro. to Discrete Structures II 4 Expectation For a rv X with pmf P{X=x}=p X (x), x Range(X), the expectation (expected value) is E[X] = x Range(X) x ¢ p X (x) • E.g., in the coin example E[X] = 3 ¢ 1/8 + 1 ¢ 3/8 + -1 ¢ 3/8 + -3 ¢ 1/8 = 0 Monday, March 23, 2009
03/22/2009 CS206 - Intro. to Discrete Structures II 5 Example 1 • Coin example X p X (x) E[X] = 0 Monday, March 23, 2009

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03/22/2009 CS206 - Intro. to Discrete Structures II 6 Example 2 A school class of 120 students are driven in 3 buses. There are 36 students in one bus, 40 in another, and 44 in the third one. When the buses arrive one student is randomly chosen. Let X be the number of students on the bus of the randomly chosen student. What is E[X]? P{X=36} = 36/120, P{X=40} = 40/120, P{X=44} = 44/120 E[X] = 36 ¢ 36/120 + 40 ¢ 40/120 + 44 ¢ 44/120 ¼ 40.27 Note: E[X] 40, the average number of students per bus Monday, March 23, 2009
03/22/2009 CS206 - Intro. to Discrete Structures II 7 Example 3 • Recall the example of tossing a coin until a head turns up. P{X=k} = p X (k) = (1-p) k-1 p, k=1,2,… E[X] = k=1 1 k p X (k) = k=1 1 kp(1-p) k-1 = p/(1-(1-p)) 2 = 1/p Note: k=0 1 kx k-1 = 1/(1-x) 2 , |x|<1 Monday, March 23, 2009

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03/22/2009 CS206 - Intro. to Discrete Structures II 8 Example 3 (cont’d) E[X] = 5/3 p=0.6 E[X] – expected number of tosses until head occurs Monday, March 23, 2009
CS206 - Intro. to Discrete Structures II 9 Example 4 – Index rv • An index rv X of an event A is defined as X = 1, if A, otherwise X = 0. What is E[X]?

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## This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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l13_expectvar - Expectation & Variance Reading: Ross, Ch...

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