The Distribution of a Random Variable

In this course, 95% of our efforts will be devoted to events that have numeric outcomes (e.g., the return on an asset, the selling price of a house, the monthly demand for a product or service, etc.).  Probabilistic events that have numerical outcomes are called random variables. 

To describe a random variable completely, we need to know two things: (1) each possible outcome; and (2) the probability of each outcome.  If we have a complete description of both, then we say we know the random variable’s distribution.  There are two general types of random variables (or distributions) encountered in this course: discrete and continuous.

A discrete random variable is one whose outcomes can be listed (like the roll of a die).  A continuous random variable is one whose outcomes are so numerous they cannot be listed.  An example of a continuous random variable would be the time elapsed between customers entering a retail establishment.  If measured to infinitesimal accuracy, one could not list all of the possibilities.  However, if we only measured elapsed times to the nearest second (or minute), then the distribution of elapsed times would be discrete.  In practice, continuous distributions are often used as approximations to discrete distributions in instances where the number of possible outcomes is so large that a continuous distribution makes the analysis easier. 

Example: A Discrete Distribution.  Define a random variable whose value is the sum of the dots obtained from rolling a pair of dice. Construct the probability distribution for this random variable.

           

Outcomes
Probabilities

                       

We frequently summarize information for a discrete random variable by means of its probability histogram.  The probability histogram is simply a visual display of the outcomes (plotted along the x-axis), and their associated probabilities, which are represented by bars (graphed along the y-axis).

Measures of a Distribution: Expectation and Variance (Discrete Case)

The expected value or mean of a discrete random variable (“r.v.” for short) X is denoted by E(X) or  μ and given by the formula μ =E(X) = Σx_ip_i

The expected value is the “theoretical” average obtained by weighting each outcome by its respective probability and then summing.  For the sum of the dice we have

            Possible values (x_i)          Probability (p_i)               Product (x_i p_i)

                        2                                  1/36                             2/36

                        3                                  2/36                             6/36

                        4                                  3/36                             12/36

                        5                                  4/36                             20/36

                        6                                  5/36                             30/36

                        7                                  6/36                             42/36

                        8                                  5/36                             40/36

                        9                                  4/36                             36/36

                        10                                3/36                             30/36

                        11                                2/36                             22/36

                        12                                1/36                             12/36

                                                                                                252/36 = 7     (μ =E(X))

Plot the value 7 on the probability histogram.  Observe that E(X) is a measure of centrality.

Example.  You sell big electric motors.  During a given week, demand for your 100-hp motor is 0, 1, or 4 (4 come on a palette).  The distribution is described below.

Demand	0	1	4
Probability	.45	.40	.15

What is your expected demand for a week?

E(X) =  (0)(.45)+(1)(.4)+(4)(.15) = 1.00

Another measure of interest is the expected value of the expression (X – E(X)²), called the variance of X, and given by the formula

Var(X) = E(X) – [E(X)]²

The variance also goes by the Greek letter σ².  The formula looks bad, but a few simple examples will clarify its calculation and help us understand what it tells us.  Note that the mean of X is needed before computing the variance.  Recall that p_i is the probability that X takes on the value x_i.