Original subtitles

-: Hello again.

In this lecture,

we are going to discuss the exponential distribution

and its main characteristics.

For starters, we define the exponential distribution

with the abbreviation Exp,

followed by a scale parameter, lambda.

We read the following statement as

Variable X follows an exponential distribution

with a scale of a half.

All right.

Variables which most closely follow an

exponential distribution are ones

with a probability that initially decreases

before eventually plateauing.

One such example is the number of views for a YouTube vlog.

There is a great interest upon release, so it starts

off with many views in the first day or two.

After most subscribers have had the chance to see the video,

the view counter slows down.

Even though the aggregate amount of views keeps increasing,

the number of new ones diminishes daily.

As time goes on, the video either becomes outdated

or the author produces new content,

so viewership focus shifts away.

Therefore, it is more likely

for a random viewing to have occurred

close to the video's initial release

than in any of the following periods.

Graphically, the PDF

of such a function would start off very high

and sharply decrease within the first few timeframes.

The curve somewhat resembles a boomerang

with each handle lining up with the X and Y axis.

All right.

We know what the PDF would look like

but what about the CDF?

In a weird way, the CDF would also resemble a boomerang.

However, this one has shifted 90 degrees to the right.

As you know,

the cumulative distribution eventually approaches one,

so that would be the value where it plateaus.

To define an exponential distribution,

we require a rate parameter,

denoted by the Greek letter, lambda.

This parameter determines how fast the CDF curve

reaches the point of plateauing,

and how spread out the graph is.

All right, let's talk

about the expected value and the variance.

The expected value for an exponential distribution

is equal to one over the rate parameter lambda.

Whilst the variance is one over lambda squared.

In data analysis,

we end up using exponential distributions quite often.

However, unlike the normal or chi squared distributions,

we do not have a table of known variables for it.

That is why sometimes we prefer to transform it.

Generally, we can take the natural logarithm

of every set of an exponential distribution

and get a normal distribution.

In statistics, we can use this new transform data

to run linear regressions.

This is one of the most common transformations

I've had to perform.

Before we move on,

we need to introduce an extremely important type

of distribution that is often used in mathematical modeling.

We're going to focus on the logistic distribution

and its main characteristics in the next video.

Thanks for watching.