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Random Walks

Random Walks. Who’s This Man?. Burton Gordon Malkiel. American economist and writer. 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.

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Random Walks

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  1. Random Walks

  2. Who’s This Man? • 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. * The path taken for each step is completely random.

  4. 4 Heads 1 Heads 3 Heads 2 Heads 0 Heads 3 Heads 2 Heads 1 Heads 0 Heads 4 Heads Heads or Tails 1 4 6 4 1

  5. 1st Coin 2nd Coin 3rd Coin 4th Coin A tree diagram of possible outcomes HHHH HHHT HHTH HHTT HTHH Number of Heads HTHT HTTH HHTT HTTT HTHT THHH HHHT HTTH HTTT THHT HHTH THHT THTT THTH HTHH THTH TTHT THTT THHH TTHH TTTH TTHH TTHT TTTH TTTT 4 3 2 1 0 Each coin shows either Heads or Tails HHHH TTTT 1 4 6 4 1

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

  7. A Binomial Random Walk Each step moves up 1 unit or down 1 unit according to the random flip of a coin 1 0 -1 2 1 1 0 0 -1 -1 -2 2 2 1 1 0 0 -1 -1 0 +2 -2 +2 +2

  8. Number of ways final value occurs Distribution of final values +4 HHHH 1 +2 4 HHTT HTHT 0 HHHT 6 HTTH HTTT HHTH THHT THTT HTHH THTH TTHT THHH TTHH TTTH -2 4 1 TTTT -4 The outcome depends upon the number of Heads, not the order in which they occur

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

  10. 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. Sunshine State Standards

  11. 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

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