VarianceDiscreteDist-4

VarianceDiscreteDist-4 - Welcome to St. Petersburg! Game...

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1 Welcome to St. Petersburg! Game set-up We have a fair coin (come up “heads” with p = 0.5) Let n = number of coin flips before first “tails” You win $2 n How much would you pay to play? Solution Let X = your winnings E[X] = I’ll let you play for $1 million. .. but just once! Takers? 0 1 3 4 2 3 1 2 0 1 2 2 1 ... 2 2 1 2 2 1 2 2 1 2 2 1 i i i 0 2 1 i Breaking Vegas Consider even money bet (e.g., bet “Red” in roulette) p = 18/38 you win $Y, otherwise (1 – p) you lose $Y Consider this algorithm for one series of bets: 1. Y = $1 2. Bet Y 3. If Win, stop 4. if Loss, Y = 2 * Y, goto 2 Let Z = winnings upon stopping E[Z] Expected winnings ≥ 0. Use algorithm infinitely often! ... ) 1 2 4 ( 38 18 38 20 ) 1 2 ( 38 18 38 20 1 38 18 2 1 38 20 1 1 38 18 38 20 38 18 2 2 38 18 38 20 0 1 1 0 i i i j j i i i Vegas Breaks You Why doesn’t everyone do this? Real games have maximum bet amounts You have finite money o Not be able to keep doubling bet beyond certain point Casinos can kick you out But, if you had: No betting limits, and Infinite money, and Could play as often as you want. .. Then, go for it!
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VarianceDiscreteDist-4 - Welcome to St. Petersburg! Game...

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