7
Lecture 3: Random Variables
Note that in the last diagram I have included the symbol X.
What is this symbol standing for?
We can
think of a statistics problem as an experiment or process with one or more possible
outcomes
.
We are not
certain which outcome will occur. For example, the next person you meet while walking down the street
may be a man or a woman. You know one of these outcomes will occur but you do not know, with
certainty, which will occur. Your computer may crash 0, 1, 2, 3, 4, or more times while writing your
econometrics project. You know one of these outcomes will occur, but there is an element of chance or
randomness involved in the process.
The
population
consists of all the possible random outcomes.
X is a symbol that stands for the
random
variable,
which we define as follows:
A
random variable
is a numerical
summary of a random outcome in a statistical or random
experiment.
A
statistical experiment
is a process leading to one or more possible outcomes with uncertainty
as to which will occur.
You sit down to write a term paper and you are undertaking a statistical experiment to determine how many
times the computer will crash. (You are also undertaking an experiment to determine what your grade will
be on the paper, but that is not the statistical problem we are interested in right now.) The number of times
your computer crashes while you are writing your term paper is random and takes on a numerical value, so
it is a random variable.
Random variables must take on numerical values.
Strictly speaking a random variable is one that can take
as outcomes only numerical values.
So if a process has as outcomes man, woman, then strictly speaking,
this is not a random variable.
But suppose we let man = 1, woman = 2. Then we do have a random
variable that can take on the value 1 or 2.
So it is always easy to define a problem in such a way that it can
be represented by the random variable.
Random variables are one of the most important concepts in statistics.
A random variable can take on
many possible values, and we do not know for certain which value will actually occur.
The set of possible
values that the random variable can take on is what we have previously referred to as the population.
So far
we have several simple examples of random variables.
1)
Let X be the gender of the next person you meet when you walk down the street. Then the
random variable X takes on the possible outcomes: 1 (male); 2 (female).
2)
Let the experiment be 'a coin is tossed'.
Then the random variable X has the possible
outcomes 1 (head), 2 (tails).
3)
Let M be the number of times your computer crashes when you write your project for this
course. The possible outcomes could be 0, 1, 2, 3, 4, 5, etc.
4)
Let the random variable be ‘ the rate you expect inflation to be next year’. The random
variable X might take on the possible values 4%, 6%, and 8% or even 5.3333333%.