Arithmetic mean of observed results = E(observed results)

[BruceZ]

This post will show that the arithmetic mean of observed results is an expected value of those observed results. This is not merely an issue of nomenclature, but follows necessarily from the precise mathematical definition of expected value. We can also compute a variance and standard deviation of observed results.

Expected value is a function of a random variable. In particular, it is a weighted sum of each possible value of the random variable multiplied by the probability that the random variable takes on that value. Thus it is a weighted average of all the values of the random variable that can be observed. Once an observation of a random variable has been made, the expected value of that random variable is simply the value of the observation itself. There is only one possible value for a random variable that has been observed, and that value is multiplied by the probability 1 to produce a constant, and that constant is the expected value of the random variable. In this way we note that the expected value of a constant is always equal to that constant. If X is a random variable, and the expected value of X is denoted E(X), then for a constant c where X=c, E(X) = E(c) = c. This fact is well known and often used.

Suppose X is a random variable which represents the amount I win in a 4 hour session. E(X) is the expected value of this random variable. X1,X2,X3 are random variables which represent what I win in 3 different 4 hour sessions. The expected value of my win for all 3 sessions is E(X1 + X2 + X3) = E(X1) + E(X2) + E(X3). EV is the expected value of my hourly rate for the 3 sessions, so EV = [ E(X1) + E(X2) + E(X3) ]/12. Now suppose I finish playing the 3 sessions, and my observed results for the 3 random variables are X1=x1, X2=x2, and X3=x3, where x1,x2,x3 are constants. Now EV = [ E(x1) + E(x2) + E(x3) ]/12 = (x1 + x2 + x3)/12. Thus my EV is simply the sum of my session results divided by the number of hours played, or the arithmetic mean of my hourly results. It is correct to refer to this as the EV of my observed results since the function EV is well defined for the constants x1,x2,x3. To suggest that this result cannot be called an EV because it is a function of observed constants would be the same as saying that 2 cannot be called the square root of 4 because the function f(x) = sqrt(x) is only a square root when applied to a variable x but not to the constant 4. Of course that would be nonsense, and this issue simply pertains to the elementary concept of functions.

By extension, we can compute a variance of observed results from the arithmetic mean (or expected value) of the squared excursions of the results from the mean. The square root of the variance of observed results is the standard deviation of observed results.