The Roll of the Dice

2021 ◽  
pp. 11-40
Author(s):  
Andrew C. A. Elliott
Keyword(s):  
Law Of Large Numbers ◽  
Gambler’S Fallacy ◽  
Long Run ◽  
Games Of Chance ◽  
Large Numbers ◽  
The Asymmetry ◽  
Gambler's Fallacy ◽  
The Law

The topic of probability is introduced through analysis of games of chance, using the casino games of roulette and the dice game craps. The nature of probability is explored, including different interpretations of what probability actually is. Ways of combining probabilities are described. The player will lose in the long run, but how long a run is needed for this to show itself? The asymmetry between the player and the gambling house is explored. The Gambler’s Fallacy is contrasted to the law of large numbers.

On the optimality of stationary replacement strategies

1980 ◽  
Vol 17 (01) ◽  
pp. 178-186 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Large Class ◽  
Law Of Large Numbers ◽  
Martingale Differences ◽  
Long Run ◽  
Large Numbers ◽  
The Law

In this paper it is shown that for a large class of replacement problems the class of stationary replacement strategies is complete, i.e. in order to minimize the average long run cost per unit time it suffices to consider replacement rules which are equal for each new unit irrespectively of what has been observed from earlier units. The main result is based on a version of the law of large numbers for martingale differences proved in the appendix.

On the optimality of stationary replacement strategies

1980 ◽  
Vol 17 (1) ◽  
pp. 178-186 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Large Class ◽  
Law Of Large Numbers ◽  
Martingale Differences ◽  
Long Run ◽  
Large Numbers ◽  
The Law

In this paper it is shown that for a large class of replacement problems the class of stationary replacement strategies is complete, i.e. in order to minimize the average long run cost per unit time it suffices to consider replacement rules which are equal for each new unit irrespectively of what has been observed from earlier units. The main result is based on a version of the law of large numbers for martingale differences proved in the appendix.

scholarly journals A new proof for the generalized law of large numbers under Choquet expectation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jing Chen ◽  
Zengjing Chen
Keyword(s):  
Probability Theory ◽  
Law Of Large Numbers ◽  
Moment Condition ◽  
Large Numbers ◽  
Choquet Expectation ◽  
The Law ◽  
Linear Probability ◽  
First Moment ◽  
Independent And Identically Distributed

Abstract In this article, we employ the elementary inequalities arising from the sub-linearity of Choquet expectation to give a new proof for the generalized law of large numbers under Choquet expectations induced by 2-alternating capacities with mild assumptions. This generalizes the Linderberg–Feller methodology for linear probability theory to Choquet expectation framework and extends the law of large numbers under Choquet expectation from the strong independent and identically distributed (iid) assumptions to the convolutional independence combined with the strengthened first moment condition.

Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio

2006 ◽  
Vol 73 (4) ◽  
pp. 673-686 ◽  
Author(s):  
M. A. Milevsky ◽  
S. D. Promislow ◽  
V. R. Young
Keyword(s):  
Mortality Risk ◽  
Law Of Large Numbers ◽  
Sharpe Ratio ◽  
Risk Premiums ◽  
Large Numbers ◽  
The Law

SYNCHRONIZATION IN RANDOMLY DRIVEN SYSTEMS

1995 ◽  
Vol 09 (16) ◽  
pp. 985-988 ◽  
Author(s):  
A.M. JAYANNAVAR
Keyword(s):  
Simple Model ◽  
Law Of Large Numbers ◽  
Driven Systems ◽  
Large Numbers ◽  
Long Time ◽  
The Law

We have solved analytically a simple model of evolution of particles driven by identical noise. We show that the trajectories of all particles collapse into a single trajectory at long time. This synchronization also leads to violation of the law of large numbers.

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