1. Random Walks

2. American economist and writer
Who’s This Man?
Burton Gordon Malkiel is best known for his top seller book on finances released in 1973, in its 9th edition
this year, A Random Walk Down Wall Street. Malkiel claimed that stock prices are totally random. There is no point tracking the trend of stocks to try to time/beat the market. Information affecting the prices of stock changes so rapidly that by the time you analyze the data, the variables impacting the price change.
Application of random walk theory: economics, genetics, physics, brain research, psychology
Burton Gordon Malkiel
American economist and writer

3. 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.
Spend awhile on this slide for it is the crux of the presentation.
* The path taken for each step is completely
random.

4. Heads or Tails 4 Heads 2 Heads 1 Heads 3 Heads 0 Heads 4 Heads 3 Heads6 4 1

5. A tree diagram of possible outcomes
1st Coin
2nd Coin
3rd Coin
4th Coin
A tree diagram of possible outcomes
HHHH
HHHT
HHTH
HHTT
HTHH
HTHT
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HTTT
THHH
THHT
THTH
THTT
TTHH
TTHT
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TTTT
Number of Heads 4 3 2 1
Each coin shows either Heads or Tails
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HTHT
HTTH
THHT
THTH
TTHH
HHHT
HHTH
HTHH
THHH
HTTT
THTT
TTHT
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HHHH
TTTT 1 4 6 4 1

6. Binomial Experiment Left or Right Heads or Tails 1 4 6 1 4 6
Repeated identical and independent steps each having exactly two choices
LLLL
RLLL
LRLL
LLRL
LLLR
RRRL
RRLR
RLRR
LRRR
RRRR
RRLL
RLRL
RLLR
LRRL
LRLR
LLRR
Left or Right
HHHH
HHHT
HHTH
HTHH
THHH
HHTT
HTHT
HTTH
THHT
THTH
TTHH
HTTT
THTT
TTHT
TTTH
TTTT
Heads or Tails 1 6 4 1 4 6 1 4 6 2 6 1 3 5 7 21 15 35 10 4 20
The distribution of possible outcomes corresponds to entries in Pascal’s triangle

7. A Binomial Random Walk
Each step moves up 1 unit or down 1 unit according to the random flip of a coin -1 1 -1 1 2 +2 -1 -2 1 -2 -1 1 2 +2 +2 -1 1 2
Head = gold coin = +1
Tail = copper coin = -1

8. Number of ways final value occurs1 4 6
Distribution of final values
HHHH +2 +4 -2 -4
The outcome depends upon the number of Heads, not the order in which they occur
HHHT
HHTH
HTHH
THHH
HHTT
HTHT
HTTH
THHT
THTH
TTHH
HTTT
THTT
TTHT
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TTTT

9. Related to Row 101 of Pascal’s Triangle
100 step random walks
Distribution of final values 2 6 1 3 5 7 21 15 35 10 4 20
Related to Row 101 of Pascal’s Triangle
Excel Link
Word Link

10. Sunshine State Standards
MA.E 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 The student uses technology, such as
graphing calculators and computer spreadsheets, to analyze
data and create graphs.
MA.E The student compares experimental results with
mathematical expectations of probabilities.

11. valcin_Rachelsummer07/quincunxmac
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

12. Drunkard’s Path GIVEN 4 CHOICES