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By Shaked M., Singpurwalla N. D.

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Bernoulli,“Specimen theoriae novae de mensura sortis,” Commentarii Academiae Scientiarum Imperalis Petropolitanea V (1738): 175–192 (translated and republished as “Exposition of a new theory on the measurement of risk,” Econometrica 22 (1954): 23–36). 40 The law of large numbers and simulation payoff of the casino is doubled. What amount must the casino require the player to stake such that, over the long term, the game will not be a losing endeavor for the casino? To answer this question, we need to calculate the expected value of the casino payoff for a single repetition of the game.

This surprising fact can be convincingly demonstrated using computer simulation. 1 describes the path of the actual number of heads turned up minus the expected number of heads when simulating 2,000 tosses of a fair coin. This process is called a random walk, based on the analogy of an indicator that moves one step higher if heads is thrown and one step lower otherwise. 1 are not exceptional. On the contrary, in fair coin-tossing experiments, it is typical to find that, as the number of tosses increases, the fluctuations in the random walk become larger and larger and a return to the zero-level becomes less and less likely.

36 36 36 3 You bet two chips each round. Thus, your average loss is 2 − 1 23 = round when you play the game over and over. 1 3 chip per Expected value and risk In the case that the random variable X is the random payoff in a game that can be repeated many times under identical conditions, the expected value of X is an informative measure on the grounds of the law of large numbers. However, the information provided by E(X ) is usually not sufficient when X is the random payoff in a nonrepeatable game.

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