c
Math 166, Fall 2015, Robert
Williams
3.1- Random Variables
A random variable is a rule that assigns one real number to each outcome of
an experiment.
There are three types of random variables:
1. A finite discrete random variable assumes only finitely many values
2. A infinite discrete random variable takes on infinitely many values, but
all possible values may be listed sequentially
3. A continuous random variable can take on any of the infinite number of
values in some interval of real numbers
Example 1 Determine the type of random variable involved in each of the following experiments:
• Rolling a die and recording the outcome
• Rolling a die 1, 000, 000 times and recording how many times a 1 appears
• Counting how many times you must flip a coin before getting heads
• Measuring the distance between a people and the nearest grocery store to
that person
• Counting how many times you can turn on a computer before it breaks
• Determining how long a light bulb can remain on until it burns out
1
c
Math 166, Fall 2015, Robert
Williams
A historgram is a graphical representation of the distribution of data. This
visual representation allows us to more easily compare data between two distributions.
The area of the region of a histogram associated with the random variable X
is P (X), the probability that X occurs. The probability that X takes on a
value Xi ≤ X ≤ Xj is equal to the total area of the histogram for the regions
corresponding to the values between Xi and Xj .
Example 2 A certain basketball player has made 4 free throws in several games.
After keeping track of his success, the coach makes a histogram as an easy way
to quickly analyze the number of successful free throws made out of the four
attempts. Using the histogram below, find the probability that, when attempting
4 free throws, the player will either make all of them or none of them.
2
c
Math 166, Fall 2015, Robert
Williams
Example 3 Two four-sided dice are rolled and the sum of the dice is recorded.
Determine the probability distribution for the experiment and draw a histogram
representing this distribution.
Example 4 A local pizza chain claims that its pepperoni pizza has more pepperoni than any of its competitors. In order to make sure they have more pepperoni,
an employee counts the number of pepperoni slices on a few randomly selected
pepperoni pizzas every day. After a few weeks, the following data is gathered:
Number of Pizzas
8
12
13
7
Number of Pepperoni Slices
59
60
62
63
Identify the random variable and display the data in a histogram.
3
c
Math 166, Fall 2015, Robert
Williams
Recall that for a sequence of n Bernoulli trials with probability of success p and
probability of failure q
k n−k
P (k) = C(n, k)p q
=
1 − p, the probability of k successes is
. This probability distribution is called the binomial
distribution.
Example 5 A class is given an exam that has 6 multiple choice questions,
each of which has 4 answer choices. Let X be the random variable that gives the
number of questions a student answers correctly. Assume that a student with a
reasonable grasp of the material has a probability of 0.8 to answer each question
correctly. Find the probability distribution for X and graph this in a historgram.
Use the histogram to determine the probability that the student gets at least 5
questions correct.
4