A random variable has a value that is uncertain
and determined by chance events. A discrete random variable
will take on values that are separate and distinct. Examples
include
 the number of airplanes owned by an airline
 the number of people waiting in line at a grocery store
checkout
 the number of cars available for rent by a rental
agency
 the number of patents developed by a research and
development department
Imagine selecting one card from a deck of cards. The value of
the card you will select is uncertain and determined by chance.
Therefore, the value of the card before you choose it is a
random variable.
The possible values this random variable can assume are the
values of each card in the deck: two, three, four, and so on. Because these
values are distinct, indivisible amounts, the random variable is
discrete.
After you have selected a card, its value is no longer uncertain and
therefore is not random. For example, if you select a five card, five
is the value that the discrete random variable has assumed. However,
five is not a random variable itself.
Probability Distributions Each possible value of a
random variable has a certain probability associated with it. For
example, when you pick a card from a deck, one of the possible values
the random variable can assume is five. The probability associated with
this value is , or approximately .08,
because 4 of the 52 cards are 5s ( simplifies to ).
The probability distribution of a discrete random variable will
tell you all of the possible values that variable may take on, as
well as the probability associated with each value.
Consider the example of a company that is interested in better
understanding the number of orders it receives each week. The
discrete random variable in this example is the number of orders per
week. The company compiles the number of orders it has received in
the past. These values, which range from 41 to 48, are the possible
values of the random variable.
The company then determines the future probability of each value
based on past data. This probability distribution is illustrated in the
table below.
In the distribution P(x = x_{i}) =
p_{i}, the following notation is used.
x = the random variable
x_{i} = the ith value of the random
variable x
p_{i} = the probability that x will equal
x_{i}
Recall from the rules of probability that the probability of an
outcome is between 0 and 1 (0 is less than or equal to
p_{i} is less than or equal to 1) and that the sum of all
probabilities must equal 1, as indicated by the notation below.
The distribution not only tells the company that the number of
orders have ranged from 41 to 48 orders per week, but also that most
weeks have between 43 and 46 orders. Another way to illustrate this
probability distribution is in the form of a graph or visual
depiction. A graph for the weekly order volumes is shown here.
This graph shows the probability of each number of weekly orders.
The height of each bar displays the probability associated with that
number of orders per week. You may notice that in the graph, it is
much easier to see the cluster of orders in the range of 43 to 46
orders per week than it was in the table of values.
Binomial Distributions
In the example
above, the probability distribution contained a number of possible
values that the random variable could assume, ranging from 41 to 48.
A binomial distribution is a special form of a discrete probability
distribution. The random variable, x, is the number of
successes in n independent trials, where each trial can
result in only two outcomes—success or failure.
