For n coin-flip betsasdescribedin Section 2.6.7, let pn denote theproportionof t between1 and n for whichthetotalwinnings Yt at time t is positive.

(a) With n = 100, simulatethisMarkovchainafewtimes,eachtimeshowingaplotof

(y1, y2, ...,y100) and reporting pn.

(b) NowsimulatethisMarkovchain100,000times.Constructahistogramofthe100,000 {pn}

valuesanddescribethedistribution.(As n increases, pn has alimitingbetadistribution

(Exercise 2.70)with α = β = 0.5, so1/2hasthelowestdensityfor pn!)