Unizor - Probability - Random Variables - Variance

Random variables take many different values with different probabilities. Full description of these discreet random variables includes enumeration of all its values and corresponding probabilities. That is a full but lengthy description of a random variable. Though it is full, it does not provide an easy answer to such question as "What is my risk if I play this game for money?" or "What is an interval I should observe the values of this random variable most of the time?" There is, however, a desire to characterize the random variable with a small number of properties to give an idea of its behavior, evaluate risk (if risk is involved) and predict its values with certain level of precision, which, arguably, is one of the main purposes of theory of probabilities.

The first such property that we have introduced is the expected value or expectation of a random variable. Though it does provide some information about the random variable, it's not really sufficient to make good predictions about its values.

Assume that our random variable ξ takes values
x1, x2,..., xN
with probabilities
p1, p2,..., pN.
Then its expectation is
E(ξ) = x1·p1+x2·p2+...+xN·pN
This expectation can be viewed as a weighted average of the values of our random variables with probabilities of these values taken as weights .

To evaluate the deviation of the values of a random variable from its expectation, we are interested in weighted average of the squares of differences between values of this random variable and its expectation, that is in the expectation of a new random variable (ξ−a)^2, where a=E(ξ):
Var(ξ) = (x1−a)^2·p1+(x2−a)^2·p2+...+(xN−a)^2·pN