Expected value, the mean of a random variable's probability distribution.

Then the values taken by the random variable are directions.

There is a probability of that this random variable will have the value 1.

Its events and random variables are the same as on the usual circle.

This set of random variables represents the risk at hand.

Another one is that the different random variables (or observations) must be independent of each other.

Since is a sum of independent -1, 1 random variables.

Recent results shows that such behavior is shared by other functions of independent random variables.

It models the state of a system with a random variable that changes through time.

Then this is done again with a new set of random variables.