Original subtitles

-: Hello again.

In this lecture

we are gonna discuss the Poisson distribution

and its main characteristics.

For starters, we denote a Poisson distribution

with the letters Po and a single value parameter, lambda.

We read the statement below as variable Y

follows a Poisson distribution with Lambda equal to four.

Okay.

The Poisson distribution deals with the frequency

with which an event occurs

within a specific interval.

Instead of the probability of an event,

The Poisson distribution requires knowing how often

it occurs for a specific period of time or distance.

For example, a firefly might light up

three times in 10 seconds on averaege.

We should use a Poisson distribution

if we want to determine the likelihood

of it lighting up eight times in 20 seconds.

The graph of the Poisson distribution plots,

the number of instances

the event occurs in a standard interval of time,

and the probability for each one.

Thus, our graph would always start from zero.

Since no event can happen a negative amount of times.

However, there is no capped the amount of times

it could occur over the time interval.

Okay. Let us explore an example.

Imagine you created an online course on probability.

Usually your students asked you around

four questions per day,

but yesterday they asked seven.

Surprised by this sudden spike

in interest from your students,

you wonder how likely it was

that they would ask exactly seven questions.

In this example, the average number of questions

you anticipate is four, so lambda equals four.

The time interval is one entire workday

and the singular instance you are interested in is seven.

Therefore, why is seven?

To answer this question

we need to explore the probability function

for this type of distribution.

All right.

As you already saw

the Poisson distribution is wildly different

from any other we have gone over so far.

It comes without much surprise

that its probability function is far different

from anything we have examined so far.

The formula looks as follows,

P of Y

equals

lambda

to the power of Y times the Euler's Number

to the power of negative lambda over y factorial.

Before we plug in the values

for our course creation example

we need to make sure that you understand the entire formula.

Let's refresh your knowledge

of the various parts of the formula.

First, the "e" you see on your screen is known

as Euler's Number or Napier's constant.

As the second name suggests

it's a fixed value approximately equal to 2.72.

We commonly observe it in physics, mathematics, and nature

but for the purpose of this example

you only need to know its value.

Secondly, a number to the power of negative n is the same

as dividing one by that number to the power of n.

In this case, e to the power of negative Lambda

is just one over e to the power of Lambda.

Right.

Going back to our example, the probability

of receiving seven questions is equal to four

raised to the seventh degree, multiplied by e

raised to the negative four over seven factorial.

That approximately equals

16,384

times 0.0183

over 5,040,

or

0.06.

Therefore, there was only a 6% chance

of receiving exactly seven questions.

So far, so good.

Knowing the probability function

we can calculate the expected value.

By definition, the expected value of Y equals the sum

of all the products of a distinct value

in this sample space and its probability.

By plugging in, we get this complicated expression.

In the additional materials attached to this lecture

you can see all the complicated algebra required to

simplify this.

Eventually we find that the expected value is simply Lambda.

Similarly, by applying the formulas we already know

the variance also ends up being equal to lambda.

Both the mean and the variance being equal to Lambda

serves as yet another example

of the elegant statistics these distributions possess

and why we can take advantage of them.

Great job everyone.

Now, if we wish to compute the probability

of an interval of a Poisson distribution

we will take the same steps we usually do

for discrete distributions.

We find the joint probability

of all individual elements within it.

You will have a chance to practice this

in the exercises after this lecture.

So far, we have discussed uniform, Bernoulli, binomial

and Poisson distributions, which are all discrete.

In the next video, we will focus

on continuous distributions and see how it differs.

Thanks for watching.