41
Stat 250 Gunderson Lecture Notes
Chapter 8: Random Variables
All models are wrong; some models are useful.
 George Box
Patterns make life easier to understand and decisions easier to make.
In Chapter 2 we
discussed the different types of data or variables and how to turn the data into useful
information with graphs and numerical summaries.
Having some notion of probability from the
previous chapter, we can now view the variables as “random variables” – the numerical
outcomes of a random circumstance.
We will look at the pattern of the distribution of the
values of a random variable and we will see how to use the pattern to find probabilities.
These
patterns will serve as models in our inference methods to come.
8.1 What is a Random Variable?
Recall in our discussion on probability we started out with some random circumstance or
experiment that gave rise to our set of all possible outcomes
S
.
We developed some rules for
calculating probabilities about various events. Often the events can be expressed in terms of a
“random variable” taking on certain outcomes. Loosely, this random variable will represent the
value of the variable or characteristic of interest, but
before we look
. Before we look, the value
of the variable is not known and could be any of the possible values with various probabilities,
hence the name of a “random” variable.
Definition:
A
random variable
assigns a number to each outcome of a random circumstance, or,
equivalently, a random variable assigns a number to each unit in a population.
We will consider
two broad classes
of random variables:
discrete
random variables and
continuous
random variables.
Definitions:
A
discrete random variable
can take one of a countable list of distinct values.
A
continuous random variable
can take any value in an interval or collection of intervals.
Try It!
Discrete or Continuous
A car is selected at random from a used car dealership lot.
For each of the following
characteristics of the car, decide whether the characteristic is a continuous or a discrete
random variable.
a. Weight of the car (in pounds).
Continuous
b. Number of seats (maximum passenger capacity).
Discrete
c. Overall condition of car (1 = good, 2 = very good, 3 = excellent).
Discrete
d. Length of car (in feet).
Continuous
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In statistics, we are interested in the
distribution of a random variable
and we will use the
distribution to compute various probabilities. The probabilities we compute (for example,
p
values in testing theories)
will help us make reasonable decisions.
So just what is the distribution of a random variable?
Loosely, it is a model that shows us what
values are possible for that particular random variable and how often those values are
expected to occur (i.e. their probabilities).
The model can be expressed as a function or table or
picture, depending on the type of variable it is.
In sections 8.2, 8.3, and 8.4 we will discuss discrete random variables and their models.
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 Winter '10
 Gunderson
 Statistics, Normal Distribution, Probability distribution, Probability theory, INDEP, George Box

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