Discrete Variables

Consider a function that can only take on a discrete set of values. Examples are flipping a coin (either heads or tails, 2 possible values), rolling a dice (6 possible values), the allowed energy levels of the hydrogen atom (En = -2.18 x 10-18J/n2). We wish to evaluate the average of this function. For starters, consider a coin flip. Assume that the probability of obtaining heads is the same as the probability for obtaining tails (hence, each has a probability of ½ of occurring). Consider the following function of the coin flip:

To see this, consider an example: The coin is flipped 1000 times, 500 times the result is heads, 500 times the result is tails. The average value of f in this example would be calculated in the standard manner:

which is just the result we started out to prove. In general, then, the average of a discrete variable is simply

where x can take on a discrete set of values (x1, x2, …), f(x) is a function of x, and p(xi) is the probability of x taking on the value xi.

Continuous Variables

Continuous variables can take on an infinite set of values. Examples include the position of a particle, the velocity of a particle, the pressure of a gas, etc. Given the above result, it is straightforward to evaluate the average of a function of a continuous variable:

where p and f are defined in a manner similar to their definitions for discrete variables.