Cumulative distribution functions model, the probability that some random variable will take on a value
less than some specific value that we set.
So we model them this way.
This is what the equation looks like.
We distinguish between probability mass functions and probability density functions and cumulative distribution
functions by using this capital F, So with probability mass and probability density functions, we
used a lowercase F.
If you see a capital F, it means that we have transitioned to a cumulative distribution function instead.
And what this says here is that the cumulative distribution function is equal to this is the probability
that our discrete random variable capital X takes on some value less than or equal to this value of
x that we set.
Now, we actually understand this better if we look at the graphs of the distributions for both discrete
and continuous random variables.
So let's start with a discrete random variable.
So we're talking about discrete random variables here.
We have already looked at the probability mass function for a discrete random variable.
This is the one we've been using throughout these lessons.
It's the probability mass function associated with rolling one six sided die one time.
And it shows us that a probability that we get a one, two, three, four, five or six when we roll
the die is always equivalent.
That's why we see that all of these bars have equal heights.
The probability is always one over six.
The probability of rolling a one is one over six, The probability of rolling a two is one over six,
etc. So this is the probability distribution showing the probability mass function for the discrete
random variable.
This graph on the right, on the other hand, is the cumulative distribution function that's associated
with this same probability mass function.
So if we were to transfer this information from the probability distribution over here to this cumulative
distribution, this is how one graph translates to the other.
In other words, this is the same probability here, the probability associated with rolling a six sided
die one time.
We're just showing the cumulative distribution.
And so what we're saying here, when we look at the graph or what this equation over here tells us is
that we can define this kind of cumulative probability.
So instead of just saying what is the probability that we roll a one or roll a four or roll a six,
this is saying what is the probability that we roll a number less than or equal to two, less than or
equal to four or less than or equal to six, for example.
So that word cumulative is appropriate because the probability accumulates.
And that idea of accumulation means that the cumulative distribution is always going to be increasing.
And in fact, because we know that probability is always defined between zero and one over here on the
left edge of the graph of our cumulative distribution function, the probability that we see is zero.
The value of the graph along the vertical axis here is at zero, and that probability is always going
to increase and increase and increase as we move to the right until eventually it gets to one.
We see here we have six over six or one, the probability tops out at one.
And so we say that the cumulative distribution has these properties, that the limit as X gets close
to negative infinity is zero, and that the limit as x gets close to positive infinity is one.
Now this limit concept comes from calculus, and that's really beyond the scope of this course.
But technically all this is really saying is that we're at a probability of zero over here on the left.
That's where this zero comes from.
And one over here on the right.
That's where this one comes from.
In other words, our cumulative distribution, the graph of our cumulative distribution is always going
to start at one and always increase and always end at positive one, which makes sense because again,
we are accumulating probability as we move to the right.
If we're rolling a six sided die one time, the probability that we roll a value less than one is of
course going to be zero, because the lowest value we can roll is a one itself.
That's why we see this zero value along the horizontal axis for these values.
Less than one these empty circles here indicate that this value is not a part of the graph, whereas
these solid filled in circles indicate that this is a part of the graph.
So what this tells us is that the probability that we roll a value less than or equal to one, we come
over to one along the horizontal axis and then we go up and find the solid part of the graph, which
is not this empty part here, but the solid part here.
We come over to the vertical axis and we see that this is one over six.
So the probability that we roll a value less than or equal to one is one sixth, the probability that
we roll a value, let's say less than or equal to three, we come over to three.
We go up until we find a solid part of the graph.
So we skip past this empty circle.
We come up here.
Then we go over along to the vertical axis.
This is at three over six.
So the probability that we roll a value less than or equal to three is three over six.
And that makes sense because less than or equal to three means we're rolling a one, a two or three,
which represents half of the total possibilities and three over six is equal to one half.
So we can see how those dice rolls are accumulating and eventually we get to x equals six here where
we can say we come up to the solid part of the graph and then we come over to the vertical axis and
we can see the probability six over six or one there, this is one.
So the probability that we roll a value less than or equal to six is of course, one.
So that's what the cumulative distribution function is looking like.
For a discrete random variable.
We are adding up all of those probabilities as we go.
It would be like taking this graph of the probability mass function and adding each of these probabilities
as we go.
So the probability that we roll a one or less is one sixth.
Then when we add another one six to that, we can say that the probability of rolling a two or less
is two over six or one third.
The probability of rolling a three or less is one sixth plus one sixth plus one sixth or three over
six or one half.
To give us yet another view on this, let's look at continuous random variables.
So it's the same idea.
We can use this cumulative distribution function for both discrete and continuous random variables.
When we have a continuous random variable, the graph of our probability density function might look
something like this, and if we transfer that over into a cumulative distribution function, we see
how this probability accumulates starting down at zero and continuing to accumulate probability all
the way up until we get to one, at which point it levels off and never rises above this value of one.
And we can imagine here that this sort of mean value in the probability density function is represented
here in the cumulative distribution function.
So we have those visuals.
If we wanted to plug into the actual function here, let's say we wanted to model the probability for
this discrete random variable representing the DI role.
We wanted to model the probability of rolling something less than or equal to four.
Then our cumulative distribution function here would be capital F and then capital X, and then we would
say of four lowercase x Equal to four is the probability that the discrete random variable x takes on
a value less than or equal to four.
If we wanted to say here for the continuous random variable, let's say that maybe this right here is
x equals two and maybe roughly the same here as something like this x equals two, Then we could do
the exact same thing for the continuous random variable and say that the cumulative distribution function
we have capital F, capital X of two is equal to the probability that the continuous random variable
represented by Capital X is less than or equal to two.
We would get that cumulative probability.
The probability that this continuous random variable is taking on any value less than two or the exact
value of two.
So anything up to and including two.
And we could find that probability on the graph by coming up from two here to the corresponding point
on the graph and then coming over here to the vertical axis.
And this value along the vertical axis, let's say that's about 0.85 based on where we put in this one
here, I'm just looking at the scale between zero and one and estimating that that's about it, 0.85.
And so maybe if this is the shape of the cumulative distribution function, then the probability that
the continuous random variable takes on a value less than or equal to two is approximately 0.85 or 85.
So again, instead of just modeling the probability of the discrete random variable, taking on a specific
value, taking on the specific value x equals three or for the continuous random variable in the probability
density function, the probability that we take on a value between, let's say one and two.
So take on some value between one and two.
That's what we did before with probability mass functions and probability density functions for discrete
and continuous random variables respectively.
Now, when we switch to cumulative distribution functions, we're looking at the probability that the
random variable takes on some value up to some specified value up to some specified value instead of
the probability of an exact value or the probability of some value in an interval.
And then the only the thing we want to say about this is that we can.
Use a cumulative distribution to find the probability that we do get a value in a particular interval.
And we use this formula here.
So if we have a cumulative distribution function, whether for a discrete random variable or a continuous
random variable, we can use the values along the cumulative distribution function to find the probability
that the random variable takes on a value in that interval.
So, for instance, going back to this discrete random variable, because we have concrete values here,
it's easy to see if we want to know the probability that when we roll the die, we get a value, let's
say between two and four.
And notice this here notation less than and less than or equal to.
So what we're saying is that we're looking for the probability that X is between two and four, not
including two, but including four.
So if we have an interval here, two, three, four, we're not including two, but we are including
four.
So the only values that would actually fall within this interval are the values three and four.
So we set that up and then we look at our cumulative distribution indicated by this capital F and we
find the value of that cumulative distribution function at B, So B is four.
We come here to four and we see that up at four.
Here we're at four over six, which is two thirds.
So we get two thirds and then we subtract the value of the cumulative distribution function evaluated
at a well, A is two.
So we come up here to two, we come up to the solid circle, come over to the vertical axis, and we
see they're two over six or one third.
So we subtract one third and the result then two over three minus one over three is one.
Third.
So this tells us that the probability that we get a value between two and four, not including two,
but including four is one third.
And of course, that makes sense because we said here that this expression showing this interval implies
getting either a three or a four.
Those are the only two values that fall within this interval.
And of course, out of six possibilities when we roll the dice by getting anywhere from a one to a six,
there are six possibilities.
There are two possibilities of those six that meet this criteria that fall within this interval.
So out of the six total possibilities, there are two that fall within the interval.
And of course that simplifies to one third, one third of our possibilities.
Meet this interval, meet this criteria.
And so the probability that we fall within this interval is one third.
So we can also use our cumulative distribution, whether it's for a discrete or continuous random variable
to calculate the probability that the random variable will take on a value in this particular interval.