Random Walks

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

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.

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

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 HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT Number of Heads 4 3 2 1 Each coin shows either Heads or Tails HHTT HTHT HTTH THHT THTH TTHH HHHT HHTH HTHH THHH HTTT THTT TTHT TTTH HHHH TTTT 1 4 6 4 1

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

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

Number of ways final value occurs 1 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 TTTH TTTT

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

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.

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

Drunkard’s Path GIVEN 4 CHOICES