Random
Walks
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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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Heads or Tails
4 Heads
1 Heads
3 Heads
4 Heads
1
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2 Heads
3 Heads
4
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2 Heads
6
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0 Heads
1 Heads
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0 Heads
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1st Coin 2nd Coin 3rd Coin
Each coin
shows either
Heads or Tails
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4th Coin
HHHH
HHHT
HHTH
HHTT
HTHH
HTHT
HTTH
HTTT
THHH
THHT
THTH
THTT
TTHH
TTHT
TTTH
TTTT
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A tree diagram of
possible outcomes
Number of Heads
4
3
2
1
HHHT
HHTH
HTHH
HHHH THHH
1
4
HHTT
HTHT
HTTH
THHT
THTH
TTHH
6
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0
HTTT
THTT
TTHT
TTTH TTTT
4
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Binomial Experiment
Repeated identical and independent
steps each having exactly two choices
Left or Right
LLRR
LRLR
LLLR LRRL
LLRL RLLR
LRLL RLRL
LLLL RLLL RRLL
1
4
6
Heads or Tails
LRRR
RLRR
RRLR
RRRL RRRR
4
HHHT
HHTH
HTHH
HHHH THHH
1
1
HHTT
HTHT
HTTH
THHT
THTH
TTHH
4
6
HTTT
THTT
TTHT
TTTH TTTT
4
1
The distribution of possible
outcomes corresponds to
entries in Pascal’s triangle
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A Binomial Random Walk
Each step moves up 1 unit or down 1 unit
according to the random flip of a coin
1
0
0
-1
2
+2
1
1
0
0
-1
-1
-2
-2
1
2
+2
0
1
-1
0
2
+2
-1
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Number of ways Distribution of
HHHH final value occurs final values
1
+4
The outcome depends
upon the number of
Heads, not the order
in which they occur
HHHT
HHTH
HTHH
THHH
+2
HHTT
HTHT
HTTH
THHT
THTH
TTHH
0
HTTT
THTT
TTHT
TTTH
-2
TTTT
-4
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4
6
4
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100 step random walks
Distribution of
final values
30
20
10
0
-10
-20
-30
-40
Related to Row 101
of Pascal’s Triangle
Excel
Link
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Word
Link
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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 analyze
data 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
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