We will now define the notion of a random variable.
Very loosely speaking, a random variable is a numerical
quantity that takes random values.
But what does this mean?
We want to be a little more precise and I'm going to
introduce the idea through an example.
Suppose that our sample space is a set of students labeled
according to their names.
Or for simplicity, let's just label them as a, b, c, and d.
Our probabilistic experiment is to pick a student at random
according to some probability law and then record their
weight in kilograms.
So for example, suppose that the outcome of the experiment
was this particular student, and the weight of
that student is 62.
Or it could be that the outcome of the experiment is
this particular student, and that particular student has a
weight of 75 kilograms.
The weight of a particular student is a number, little w.
But let us think of the abstract concept of weight,
something that we will denote by capital W. Weight is an
object whose value is determined once you tell me
the outcome of the experiment, once you tell me which student
was picked.
In this sense, weight is really a function of the
outcome of the experiment.
So think of weight as an abstract box that takes as
input a student and produces a number, little w, which is the
weight of that particular student.
Or more concretely, think of weight with a capital W as a
procedure that takes a student, puts him or her on a
scale, and reports the result.
In this sense, weight is an object of the same kind as the
square root function that's sitting inside your computer.
The square root function is a function.
It's a subroutine, perhaps it is a piece of code, that takes
as input a number, let's say the number 9, and produces
another number.
In this case, it would be the number 3, which is the
square root of 9.
Notice here the distinction that we will keep emphasizing
over and over.
Square root of 9 is a number.
It is the number 3.
The box square root is a function.
Now, let us go back to our probabilistic experiment.
Note that a probabilistic experiment such as the one in
our example can have several associated random variables.
For example, we could have another random variable
denoted by capital H, which is the height of a student
recorded in meters.
So if the outcome of the experiment, for example, was
student a, then this random variable would take a value
which is the height of that student, let's say it was 1.7.
Or if the outcome of the experiment was student c, then
we would record the height of that student.
And let's say it turns out to be 1.8.
Once again, height with a capital H is an abstract
object, a function whose value is determined once you tell me
the outcome of the experiment.
Now, given some random variables, we can create new
random variables as functions of the
original random variables.
For example, consider the quantity defined as weight
divided by height squared.
This quantity is the so-called body mass index, and it is
also a function on the sample space.
Why is it a function on the sample space?
Well, because once an outcome of the experiment is
determined, suppose that the outcome of the experiment was
the blue student, then these two numbers, 62 and 1.7, are
also determined.
And using those numbers, we can carry out this calculation
and find the body mass index of that particular student,
which in this case would be 21.5.
Or if it happened that this student was selected, then the
body mass index would turn out to be some other number.
In this case, it would be 23.
So again, we see that the body mass index can be viewed as an
abstract concept defined by this formula.
But once an outcome is determined, then the body mass
index is also determined.
And so the body mass index is really a function of which
particular outcome was selected.
Let us now abstract from the previous discussion.
We have seen that random variables are abstract objects
that associate a specific value, a particular number, to
any particular outcome of a probabilistic experiment.
So in that sense, random variables are functions from
the sample space to the real numbers.
They are numerical functions, but as numerical functions
they can either take discrete values, for example the
integers, or they can take continuous values, let's say
on the real line.
For example, if your random variable is the number of
heads in 10 consecutive coin tosses, this is a discrete
random variable that takes values in the
set from 0 to 10.
If your random variable is a measurement of the time at
which something happened, and if your timer has infinite
accuracy, then the timer reports a real number and we
would have a continuous random variable.
In this lecture sequence and in the next few ones, we will
concentrate on discrete random variables because they are
easier to handle.
And then later on, we will move to a discussion of
continuous random variables.
Throughout, we want to keep noting this very important
distinction that I already brought in the discussion for
a particular example, but it needs to be emphasized and
re-emphasized.
That we make a distinction between random variables,
which are abstract objects.
They are functions on the sample space and they are
denoted by uppercase letters.
In contrast, we will use lower case letters to indicate
numerical values of the random variables.
So little x is always a real number, as opposed to the
random variable, which is a function.
One point that we made earlier is that for the same
probabilistic experiment we can have several random
variables associated with that experiment.
And we can also combine random variables to
form new random variables.
In general, a function of random variables has numerical
values that are determined by the numerical values of the
original random variables.
And so, ultimately, they are determined by the outcome of
the experiment.
So a function of random variables has a numerical
value which is completely determined by the outcome of
the experiment.
And so a function of random variables is
also a random variable.
As an example, we could think of two random variables, X and
Y, associated with the same probabilistic experiment.
And then define a random variable, let's say X plus Y.
What does that mean?
X plus Y is a random variable that takes the value little x
plus little y when the random variable capital X takes the
value little x and capital Y takes the value little y.
So X and Y are random variables.
X plus Y is another random variable.
X and Y will take numerical values once the outcome of the
experiment has been obtained.
And if the numerical values that they take are little x
and little y, then the random variable X plus Y will take
the numerical value little x plus little y.
So we can now move on and start doing some interesting
things about random variables.
Characterize them, describe them, give some examples, and
introduce some new concepts associated with them.