random walks

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C A LC A LRandom WalksRandom WalksExitExit AnimationsAnimations SlideSlide 5.31/10

Random Random WalksWalks

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Who’s This Man?

• Burton Gordon Malkiel•American economist and writer

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What is the idea behind random walk theory?

* Random walk also called drunkard’s walk is based on the premise that one takes a number of successive steps.

* Each step is of equal length.

* Each step is independent from another.

* Each step has two paths of equal likelihood.

* The path taken for each step is completely random.

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11 44 66 44 11

4 Heads4 Heads

3 Heads3 Heads 2 Heads2 Heads

1 Heads1 Heads

0 Heads0 Heads

4 Heads4 Heads 3 Heads3 Heads 2 Heads2 Heads 1 Heads1 Heads 0 Heads0 Heads

Heads or TailsHeads or Tails

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1st Coin1st Coin 2nd Coin2nd Coin 3rd Coin3rd Coin 4th Coin4th Coin

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Each coin Each coin shows either shows either


44 0033 22 11Number of HeadsNumber of Heads

A tree diagram of A tree diagram of possible outcomespossible outcomes

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22

66 66

1111 11

33 331111

111111

11 55

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353511

1155 11

2121 77111515

113535

66

1010 10104444

2020

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The distribution of possible The distribution of possible outcomes corresponds to outcomes corresponds to entries in Pascal’s triangleentries in Pascal’s triangle

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LLLLLLLL RLLLRLLLLRLLLRLLLLRLLLRLLLLRLLLR

RRRLRRRLRRLRRRLRRLRRRLRRLRRRLRRR

RRRRRRRRRRLLRRLLRLRLRLRLRLLRRLLRLRRLLRRLLRLRLRLRLLRRLLRR

Left or RightLeft or Right

Binomial ExperimentBinomial ExperimentRepeated identical and independent Repeated identical and independent steps each having exactly two choicessteps each having exactly two choices

11 1144 66 44 11 1144 66 44

11 1166 4444

11 1144 66 44 11 1144 66 44

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A Binomial Random WalkA Binomial Random WalkEach step moves up 1 unit or down 1 unit Each step moves up 1 unit or down 1 unit according to the random flip of a coinaccording to the random flip of a coin

00

-1-1

11

22

00

-1-1

11

22

00

-1-1

11

-1-1

-2-2

00

11

00

-1-1

11

22

+2+2

+2+2+2+2

-2-2

00

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00

+2+2

+4+4

-2-2

-4-4

Number of ways Number of ways final value occursfinal value occurs

11

44

66

44

11

Distribution of Distribution of final valuesfinal values

The outcome depends The outcome depends upon the number of upon the number of Heads, not the order Heads, not the order in which they occurin which they occur

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-40

-30

-20

-10

0

10

20

30 Distribution of Distribution of final valuesfinal values

Excel Excel LinkLink

Word Word LinkLink

100 step random walks100 step random walks

22

66 66

1111 11

33 331111

111111

11 55

77 2121151511

353511

1155 11

2121 77111515

113535

66

1010 10104444

2020

11

Related to Row 101 Related to Row 101 of Pascal’s Triangleof Pascal’s Triangle

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Sunshine State Standards

MA.E.1.3.1 The student collects, organizes, and displays

data in a variety of forms, including tables, line graphs,

charts, bar graphs,to determine how different ways of presenting data can lead to different interpretations.

MA.E.1.3.3 The student uses technology, such as graphing calculators and computer spreadsheets, to

analyzedata and create graphs.

MA.E.2.3.1 The student compares experimental results with

mathematical expectations of probabilities.

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Food for Thought…

1. Why is random walk theory also called the gambler’s ruin theory?

2. Will a drunkard ever get back to his home from a city bar? Explain.

3. Would a “drunk” bird ever find its nest? Explain.

valcin_Rachelsummer07/quincunxmac

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Drunkard’s Path GIVEN 4 CHOICES

random walks

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