Monday, September 8, 2014

Unizor - Probability - Random Variables - Expected Values

Assume our random experiment (like spinning the roulette wheel and a ball on it) results in the following K elementary events (like ball stops in a partition with some number):
Ω = { e1, e2,..., eK }
Assume further that we know the probabilities of each elementary event (maybe, they have equal chances, maybe not):
P(ei) = pi (i runs from 1 to K)
Finally, assume we have defined a random variable on this sample space that represents a numerical result of an experiment (like winning on a $1 bet):
ξ(ei) = xi (i runs from 1 to K)
If we conduct our experiment N times (consider this a very large number), we expect, approximately, that elementary event e1 will occur in N·p1 number of cases with our random variable, correspondingly, taking a value of x1.
Similarly, in, approximately, N·p2 number of cases our random experiment will end up at elementary event e2 and our random variable will take a value of x2.
And similar for all other indices.
Knowing this statistics, we can approximate the average value of our random variable ξ. If during N experiments in, approximately, N·p1 number of cases it took value x1, in, approximately, N·p2 number of cases it took value x2, etc. then the sum of all values in took in all experiments equals to
N·p1·x1 + N·p2·x2 + ... + N·pK·xK
and the average value of our random variable ξ per single experiment equals to
E(ξ) = p1·x1+p2·x2+...+pK·xK
This value depends only on probabilities of elementary events pi and corresponding values of our random variable xi and is called mathematical expectation of a random variable or just expectation or expected value.

Can we say that our random variable "takes an average value" calculated above as E(ξ)? No, this is not correct. It might never take this value. But, with the number of random experiments, where it is a numerical result, grows to infinity, its average value per experiment (that is, a sum of all values divided by the number of experiments) tends to E(ξ). The correct statement about this is that our random variable has an expected value (calculated above based on probabilities and individual values) E(ξ).

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