A random variable can take different numerical values
depending on the outcome of the experiment.
Some outcomes are more likely than others, and similarly
some of the possible numerical values of a random variable
will be more likely than others.
We restrict ourselves to discrete random variables, and
we will describe these relative likelihoods in terms
of the so-called probability mass function, or PMF for
short, which gives the probability of the different
possible numerical values.
The PMF is also sometimes called the probability law or
the probability distribution of a discrete random variable.
Let me illustrate the idea in terms of a simple example.
We have a probabilistic experiment with
four possible outcomes.
We also have a probability law on the sample space.
And to keep things simple, we assume that all four outcomes
in our sample space are equally likely.
We then introduce a random variable that associates a
number with each possible outcome as
shown in this diagram.
The random variable, X, can take one of
three possible values--
namely 3, 4, or 5.
Let us focus on one of those numbers--
let's say the number 5.
So let us focus on x being equal to 5.
We can think of the event that X is equal to 5.
Which event is this?
This is the event that the outcome of the experiment led
to the random variable taking a value of 5.
So it is this particular event which consists of two
elements, namely a and b.
More formally, the event that we're talking about is the set
of all outcomes for which the value, the numerical value of
our random variable, which is a function of the outcome,
that numerical value happens to be equal to 5.
And in this example it is a set
consisting of two elements.
It's a subset of the sample space.
So it is an event.
And it has a probability.
And that probability we will be
denoting with this notation.
And in our case this probability is equal to 1/2.
Because we have two outcomes, each one has probability 1/4.
The probability of this event is equal to 1/2.
More generally, we will be using this notation to denote
the probability of the event that the random variable, X ,
takes on a particular value, x.
This is just a piece of notation, not a new concept.
We're dealing with a probability, and we indicate
it using this particular notation.
More formally, the probability that we're dealing with is the
probability, the total probability, of all outcomes
for which the numerical value of our random variable is this
particular number, x.
A few things to notice.
We use a subscript, X, to indicate which random variable
we're talking about.
This will be useful if we have several
random variables involved.
For example, if we have another random variable on the
same sample space, Y, then it would have its own probability
mass function which would be denoted with this particular
notation here.
The argument of the PMF, which is x, ranges over the possible
values of the random variable, X. So in this sense, here
we're really dealing with a function.
A function that we could denote just by p with a
subscript x.
This is a function as opposed to the specific
values of this function.
And we can produce plots of this function.
In this particular example that we're dealing with, the
interesting values of x are 3, 4, and 5.
And the associated probabilities are the value of
5 is obtained with probability 1/2, the value of 4--
this is the event that the outcome is c, which has
probability 1/4.
And the value of 3 is also obtained with probability 1/4
because the value of 3 is obtained when the outcome is
d, and that outcome has probability 1/4.
So the probability mass function is a function of an
argument x.
And for any x, it specifies the probability that the
random variable takes on this particular value.
A few more things to notice.
The probability mass function is always non-negative, since
we're talking about probabilities and
probabilities are always non-negative.
In addition, since the total probability of all outcomes is
equal to 1, the probabilities of the different possible
values of the random variable should also add to 1.
So when you add over all possible values of x, the sum
of the associated probabilities
should be equal to 1.
In terms of our picture, the event that x is equal to 3,
which is this subset of the sample space, the event that x
is equal to 4, which is this subset of the sample space,
and the event that x is equal to 5, which is this subset of
the sample space.
These three events--
the red, green, and blue--
they are disjoint, and together they cover the entire
sample space.
So their probabilities should add to 1.
And the probabilities of these events are the probabilities
of the different values of the random variable, X. So the
probabilities of these different values
should also add to 1.
Let us now go through a simple example to illustrate the
general method for calculating the PMF of a
discrete random variable.
We will revisit our familiar example involving two rolls of
the tetrahedral die.
And let X be the result of the first roll, Y be the result of
the second roll.
And notice that we're using uppercase letters.
And this is because X and Y are random variables.
In order to do any probability calculations, we also need the
probability law.
So to keep things simple, let us assume that every possible
outcome, there's 16 of them, has the same probability which
is therefore 1 over 16 for each one of the outcomes.
We will concentrate on a particular random variable
defined to be the sum of the random variables, X and Y. So
if X and Y both happen to be 1, then Z will take
the value of 2.
If X is 2 and Y is 1 our random variable will take the
value of 3.
And similarly if we have this outcome, in those outcomes
here, the random variable takes the value of 4.
And we can continue this way by marking, for each
particular outcome, the corresponding value of the
random variable of interest.
What we want to do now is to calculate the PMF of this
random variable.
What does it mean to calculate the PMF?
We need to find this value for all choices of z, that is for
all possible values in the range of our random variable.
The way we're going to do it is to consider each possible
value of z, one at a time, and for any particular value find
out what are the outcomes--
the elements of the sample space--
for which our random variable takes on the specific value,
and add the probabilities of those outcomes.
So to illustrate this process, let us calculate the value of
the PMF for z equal to 2.
This is by definition the probability that our random
variable takes the value of 2.
And this is an event that can only happen here.
It corresponds to only one element of the sample space,
which has probability 1 over 16.
We can continue the same way for other values of z.
So for example, the value of PMF at z equal to 3, this is
the probability that our random variable takes the
value of 3.
This is an event that can happen in two ways--
it corresponds to two outcomes--
and so it has probability 2 over 16.
Continuing similarly, the probability that our random
variable takes the value of 4 is equal to 3 over 16.
And we can continue this way and calculate the remaining
entries of our PMF.
After you are done, you end up with a table--
or rather a graph--
a plot that has this form.
And these are the values of the different probabilities
that we have computed.
And you can continue with the other values.
It's a reasonable guess that this was going to be 4 over
16, this is going to be 3 over 16, 2 over 16, and 1 over 16.
So we have completely determined the PMF of our
random variable.
We have given the form of the answers.
And it's always convenient to also provide a plot with the
answers that we have.