# Absolutely Fair

1. Dec 22, 2012

### Biosyn

What does the term 'absolutely fair' in statistics & probability mean?

Does it mean that the probability for each variable* is 0.
And that the expected value is 0.

http://mathworld.wolfram.com/AbsolutelyFair.html

Last edited: Dec 22, 2012
2. Dec 23, 2012

### Stephen Tashi

It doesn't mean that because it doesn't make sense to say "the probability for each variable is 0". Random variables don't have probabilities. It is their possible values that have probabilities.

3. Dec 23, 2012

### chiro

Hey Biosyn.

To add to Stephen Tashi's comments, I think you should consider the context of what the random variable is in.

One way of assessing "fair" is that probabilities are equal in a random variable. If this is discrete uniform with N states then P(X = x) = 1/N can be considered "fair" as can a continuous uniform with P(X = x) = 1/N in the interval [a,a+N].

In finance, we consider fair to be one where there is no risk of arbitrage or a "free lunch" under specific assumptions (which may not be right in practice).

We also consider fairness in the context of unpredictability or pure-randomness where no information at all helps predict the outcome and this is a case of maximum entropy which when used in the proper context (i.e. for discrete random variables with a finite number of outcomes) yields the uniform distribution in discrete state-space.

4. Dec 25, 2012

### lavinia

In finance one has the idea of a fair game. The naive model is a random walk or a Brownian motion but more generally the model is called a martingale. This is a game where the current value of a random variable(e.g. the price of a security) gives you the best estimate of its future value after you have accounted for all of the relevant information about it.

This means that any change in value e.g. change in the price of a stock is completely unpredicatable. The game would be unfair if some players had special information that was not generally available that enabled them to get an edge in the betting.For instance insider trading or knowlege of price patterns that were difficult to find and not generally seen by the marketplace.

Last edited: Dec 25, 2012
5. Dec 26, 2012

### Stephen Tashi

which says:

That definition is expressed in terms of "expectations" of random variables rather than probabilities. It has a misprint since the first condition should say $<X_n> = 0$ instead of $<X_1> = 0$. The notation $<X_n>$ refers to the expected value of $X_n$.

This definition differs from the definition of a martingale.

6. Dec 26, 2012

### lavinia

Steve it seems to me that this definition is the same as a type of Martingale. If one takes the new process which is the sum of the X's then the expectation of the sum in the next period is its current value.

7. Dec 27, 2012

### lavinia

If a betting game is fair them your current winnings should be your expected future winnings.