The last discrete random
variable that we will discuss
is the so-called geometric
random variable.
It shows up in the context of
the following experiment.
We have a coin and we toss it
infinitely many times and
independently.
And at each coin toss we have a
fixed probability of heads,
which is some given number, p.
This is a parameter that
specifies the experiment.
When we say that the infinitely
many tosses are
independent, what we mean in a
mathematical and formal sense
is that any finite subset of
those tosses are independent
of each other.
I'm only making this comment
because we introduced a
definition of independence of
finitely many events, but had
never defined the notion of
independence or infinitely
many events.
The sample space for this
experiment is the set of
infinite sequences of
heads and tails.
So a typical outcome
of this experiment
might look like this.
It's a sequence of heads and
tails in some arbitrary order.
And of course, it's an infinite
sequence, so it
continues forever.
But I'm only showing
you here the
beginning of that sequence.
We're interested in the
following random variable, X,
which is the number of tosses
until the first heads.
So if our sequence looked like
this, our random variable
would be taking a value of 5.
A random variable of this kind
appears in many applications
and many real world contexts.
In general, it models situations
where we're waiting
for something to happen.
Suppose that we keep doing
trials at each time and the
trial can result either
in success or failure.
And we're counting the number
of trials it takes until a
success is observed for
the first time.
Now, these trials could be
experiments of some kind,
could be processes of some kind,
or they could be whether
a customer shows up in a store
in a particular second or not.
So there are many diverse
interpretations of the words
trial and of the word success
that would allow us to apply
this particular model to
a given situation.
Now, let us move to the
calculation of the PMF of this
random variable.
By definition, what we need to
calculate is the probability
that the random variable
takes on a
particular numerical value.
What does it mean for
X to be equal to k?
What it means is that the first
heads was observed in
the k-th trial, which means
that the first k minus 1
trials were tails, and then were
followed by heads in the
k-th trial.
This is an event that only
concerns the first k trials,
and the probability of this
event can be calculated using
the fact that different coin
tosses or different trials are
independent.
It is the probability of tails
in the first coin toss times
the probability of tails in the
second coin toss, and so
on, k minus 1 times.
So we get an exponent here
of k minus 1 times the
probability of heads in
the k-th coin toss.
So this is the form of the PMF
of this particular random
variable, and that formula
applies for the possible
values of k, which are the
positive integers.
Because the time of the
first head can only
be a positive integer.
And any positive integer is
possible, so our random
variable takes values in a
discrete but infinite set.
The geometric PMF has a
shape of this type.
Here we see the plot for the
case where p equals to 1/3.
The probability that the first
head shows up in the first
trial is equal to p, that's
the probability of heads.
The probability that it shows up
in the next trial, that the
first head appears in the second
trial, this is the
probability that we had heads
following a tail.
So we have the probability of
a tail and then times the
probability of a head.
And then each time that we move
to a further entry, we
multiply by a further
factor of 1 minus p.
Finally, one little
technical remark.
There's a possible and rather
annoying outcome of this
experiment, which would be that
we observe a sequence of
tails forever and no heads.
In that case, our random
variable is not well-defined,
because there is no first
heads to consider.
You might say that in this case
our random variable takes
a value of infinity, but we
would rather not have to deal
with random variables that
could be infinite.
Fortunately, it turns out that
this particular event has 0
probability of occurring, which
I will now try to show.
So this is the event that
we always see tails.
Let us compare it with the event
where we see tails in
the first k trials.
How do these two
events relate?
If we have always tails, then
we will have tails in the
first k trials.
So this event implies
that event.
This event is smaller
than that event.
So the probability of this event
is less than or equal to
the probability of that
second event.
And the probability of that
second event is 1
minus p to the k.
Now, this is true no matter
what k we choose.
And by taking k arbitrarily
large, this number here
becomes arbitrarily small.
Why does it become arbitrarily
small?
Well, we're assuming that p is
positive, so 1 minus p is a
number less than 1.
And when we multiply a number
strictly less than 1 by itself
over and over, we get
arbitrarily small numbers.
So the probability of never
seeing a head is less than or
equal to an arbitrarily
small positive number.
So the only possibility for this
is that it is equal to 0.
So the probability of not ever
seeing any heads is equal to
0, and this means that
we can ignore
this particular outcome.
And as a side consequence
of this, the sum of the
probabilities of the different
possible values of k is going
to be equal to 1, because we're
certain that the random
variable is going to take
a finite value.
And so when we sum probabilities
of all the
possible finite values,
that sum will have
to be equal to 1.
And indeed, you can use the
formula for the geometric
series to verify that, indeed,
the sum of these numbers here,
when you add over all values of
k, is, indeed, equal to 1.
