Math 207—Discrete Random Variables (General Information)

A random variable is a variable whose value is a numerical outcome of a random experiment. A discrete random variable takes on at most a countable number of possible values. And the probability distribution of a discrete random variable, X, gives all the possible values of X along with the associated probabilities. That is, gives for all possible values of x (note the probabilities must sum to 1).

As an example, suspend reality and imagine a fair, 3-sided die. Now suppose this die is rolled twice. Let X be the sum of the numbers on the two rolls. (Note that X is a discrete random variable.) Determine the probability distribution of X.

First write out the sample space for this experiment:

S = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

Note the random variable X “maps” the outcome (1,1) to a numerical value of 2; the outcomes (1,2) and (2,1) to a numerical value of 3; the outcomes (1,3), (2,2), and (3,1) to a numerical value of 4; the outcomes (2,3) and (3,2) to a numerical value of 5; and the outcome (3,3) to a numerical value of 6.

Because the die is fair, each simple outcome has probability . Then the probability distribution of X is

Value of X	2	3	4	5	6
Probability	1/9	2/9	3/9	2/9	1/9

In a graph, the probability distribution is

Just as with sample distributions, we can discuss the center and spread of probability distributions. The expected value of a discrete random variable, X, is defined as , where the summation is taken over all possible x values.

· The expected value is also called the mean. And the notations E(X) and are interchangeable (as are the terminologies expected value and mean). Recall the sample mean was denoted ; yet the mean of a probability distribution (or of an entire population) is denoted .

· The mean is simply a weighted average of the possible values of X, where each value is “weighted” by its associated probability. Note the mean is still the balance point of the distribution.

· The mean is not necessarily a possible value from the experiment. Conceptually, you can think of it as the long-run sample average value of X based on many, many runs of the experiment.

· In our example, it’s easy to see visually the mean is 4. To check this using the formula: .

The variance of a discrete random variable is a particular expectation: . This can be determined using the formula , where the summation is taken over all possible x values.

· The variance of a probability distribution (or of an entire population) is denoted as (recall the variance of a sample of data points is denoted ).

· The variance is simply a weighted average, where each squared distance from the mean is “weighted” by its corresponding probability. The variance still measures the spread of the x values around the mean.

· The standard deviation of a probability distribution is simply the square root of the variance (hence, the standard deviation is back in the original units). The standard deviation is denoted .

· In our example, the variance is And the standard deviation is 15.