- Today I'll be introducing the exponential distribution,
which is an exciting continuous distribution
that's closely related
to the Poisson distribution or Poisson process.
Recall that a Poisson process is
where we have a continuous interval like distance here,
and we have discrete events occurring
in that continuous interval.
And the example that I talked about before,
we have a car and is traveling along
and we've got potholes.
Potholes are discrete events that occur
in a continuous interval.
If we zoomed in on any sub-interval,
regardless of how small it is,
you can potentially have two occurrences
or events that are occurring right next
to each other in time.
So the Poisson process is when you have discrete events
that are just randomly occurring in some interval,
a continuous interval.
The distribution function
for the Poisson distribution is given down here.
X here represents discrete events.
So this answers the question of,
"What's the probability of getting X number of events
in a continuous interval?"
The mean is lambda times T,
T is the period of interest.
You can plug that in here
and you can estimate the probabilities
of a certain number of events occurring in an interval.
Using the Poisson distribution,
we answered questions like,
"What is the probability
that there will be exactly six earthquakes
in the next two years?"
And what we did was we calculated a background average rate
of occurrence, 200 earthquakes or events per 40 years
and that's five per year.
If we wanted to answer this question,
X would equal six, exactly six events or earthquakes.
Our time interval would be two years.
Lambda T, which is also known as the mean,
is equal to, in this case, five times two,
and that would be 10.
You could plug that into this function with X equals six.
And when you do this, you would get about 6.3%.
There's a 6.3% chance that in the next two years,
there's gonna be exactly six earthquakes.
Now what we're gonna do is we're gonna ask questions like,
"What's the probability that there will be
at least three months until the next earthquake?"
This is known as the exponential distribution,
and X represents a continuous interval.
In the Poisson distribution,
X represented discreet number of events,
but in the exponential distribution,
X is going to represent a continuous interval.
So we're kind of flipping the Poisson distribution
on its head, and we're saying now, instead of
what's the probability of getting a number of events
in an interval, we're asking,
what's the probability of an interval between events.
In other words, if we were to look at earthquakes,
the exponential distribution tells us probabilities related
to the lengths in time, in this case,
if this is earthquakes, between events,
and importantly it's always the interval
between subsequent events.
So the exponential distribution describes distance
or length, area, volume or time,
between events described by a Poisson process,
discrete events occurring on a continuous interval.
The random variable X,
in the exponential distribution,
is the interval between two events.
It is a continuous random variable,
and can be described by this function.
So this is just the little F of X, the non cumulative.
Oftentimes we are interested
in the cumulative exponential distribution.
So if you integrated this,
if you integrated little F of X,
you would get capital F of X down here,
one minus E to the negative lambda X.
The mean of the exponential distribution is one over lambda,
and the variance is one over lambda squared.
It's a lot easier for you guys to learn,
if I go through an example.
200 earthquakes in 40 years, what's the probability
that there will be at least three months
until the next earthquake?
X then is going to be three,
and it's not exactly three, it's at least,
which means that we're gonna be looking
for the probability that X is greater than
or equal to three.
In other words, if we were to plot X,
which is time and our probability density function
for the exponential distribution,
the exponential distribution looks something like this.
What we're interested in is three months.
What's the probability that the next events
will occur at least three months from now?
And so that's represented here by that shaded region.
It could occur right at 3.1 months,
but it could also occur 10 months,
or 100 months from now.
So we have to sum the area underneath
that curve greater than three.
So what we're trying to find is the probability
that X is greater than
or equal to three.
In terms of cumulative probabilities,
how would we write the probability that X is greater than
or equal to three?
This is the same as taking one minus the cumulative
up to three, F of three represents this region here
on the left.
If we subtract that from one,
because the area underneath a probability density function
is always gonna be one,
then this will tell us what we're looking for.
F of X is equal to one minus E to the negative lambda X,
lambda is going to be 200 divided by 40,
which is five per year.
We wanna make sure, cause X down here is in months,
we wanna make sure that we convert this to months,
so we can multiply one year per 12 months
and really lambda is equal to five over 12.
When we plug this into the cumulative function down here,
when I substitute capital F of three into here,
and I subtract from one, these ones will actually cancel.
And what we're end up with
is the probability that X is greater than
or equal to three is just E to the negative lambda X,
lambda is five twelfths,
our X value is three,
and when we do that, we get the probability
that the next earthquake is gonna be greater
than three months from now to be about 0.287.
Let me show you how we can do this in Excel.
What I'm gonna do first is I'm just gonna put in
a cell here for lambda,
and that equals five divided by 12 and that's per month.
Our X value is equal to three months,
and we're trying to calculate greater than three months.
So I can use the expon.dist function in Excel.
The X value of interest is three, lambda is up here,
I've calculated five over 12, and I wanna do cumulative.
So this would give me capital F of three, up to three.
But in order to get greater than three,
I have to subtract that from one,
and we get the same result that I did using a calculator.
Let's go through one more quick example.
Same scenario.
What's the probability that the next earthquake will occur
between two and five months from now?
Again, we have X in months, we have little F of X here.
The exponential distribution looks something like this.
And we're being asked, what's the probability
that starting now the next earthquake is going to occur
between two and five months,
which represents this shaded region.
How would we determine that probability
in terms of the cumulative function, capital F of X?
We could take capital F of five,
and subtract capital F of two,
and that will give us the probability
that we are looking for.
And so I can type this in here,
the exponential distribution of five months
with our lambda up here.
I want to be cumulative capital F.
I do the same thing, but X equals two.
And when we do that,
and we get about a 31% chance
that the next earthquake will occur
between two and five months from now.
Thanks for watching this screen cast
on the exponential distribution.