All language subtitles for 015 A Practical Example of Probability Distributions_en

Instructor: Welcome back.

In this practical example

we will explore several scenarios

where understanding how a data set is distributed

is truly beneficial.

We will examine different data samples

which follow a normal, a Student's T,

and a Poisson distribution.

Furthermore, we will analyze instances

of exponential and binomial data

to help us appreciate the elegant statistics

these distributions possess.

Let's begin.

Imagine you are working as a head project manager

for one of the most renowned companies

in the world of video games, EA Games.

Your various responsibilities include:

supervising the development and release of the 2018 edition

of the soccer game titled "FIFA 19".

Above all else, you need to ensure the game is well rounded

and provides a genuinely enjoyable experience

for all the customers.

The game has a professional competitive scene

so it needs to be balanced.

By balanced, we mean that no team or individual player

should invariably be a preferred option

regardless of the opposition.

Therefore, we expect to have an equal number

of good players and poor players in the game.

Let's see if that's the case.

We provided you with access to a data set

containing the stats for each individual player

in "FIFA 19".

So let's have a closer look.

You can use Microsoft Excel to open the FIFA 19 file

accompanying this lecture.

To begin with, examine the overall column.

It represents the quality of a player

in their natural position on a scale from one to 100.

This value is a sort of weighted average

of the many individual stats each player has.

As you probably know, the importance of attributes varies

for different positions on the field.

For instance, acceleration and top speed

are more important for a winger than tackling.

However, the inverse is true for center backs.

Thus, we alter the weight for each stat

based on the position of the player.

Therefore, we do not have a single formula

which calculates the overall evaluation.

To get an idea of how well distributed

the overall values are,

we can construct a histogram and set the bin size to one.

We do so by selecting the overall column,

clicking on insert,

then insert statistics chart and selecting histogram.

To adjust the size of the bins,

right click on the x-axis of the graph

and press format axis before setting bandwidth to one.

The graph is bell-shaped

and resembles a normal distribution.

But wait, aren't we dealing with discrete values?

How can this be a normal distribution?

Although, although that may be true

continuous variables can take discreet values

but not vice versa.

Furthermore, since we are dealing with rounded averages

we are inclined to believe

that the overall value is not entirely discreet

but rather an approximation.

Let's take a closer look at the graph.

Now we can notice its thin tails

which suggest a smaller number of outliers.

This reflects real life quite accurately

since very few professional players are exceptionally good

or bad at every single aspect of the sport.

Besides even the least skilled professional soccer players

are far superior to the average person.

That explains why the lowest overall values

start from around 50 rather than zero.

The stats should reflect the performance of players

in the real world.

As normal distribution is the most frequently observed

in nature, it is only logical

that the data resembles this distribution.

Moreover, the bell-shaped graph with thin tails

further supports this idea.

Since one of the main characteristics

of a normal distribution is symmetry,

the overall values are symmetrically distributed.

Thus, we can safely consider the game balanced

and acceptable for competitive play.

It is also worth noting that players

within the single team or division share similar stats.

This skews the data a certain way

and explains why we cannot expect the values

to follow a normal distribution.

Now if we wish to further test the balance

of the overall stats,

we can examine a small sample of random players.

For instance, we can construct a histogram

of the first 30 players in the data set

based on their ID number.

Since our data is limited

we need to adjust the size of the bins,

otherwise it is possible for each value to occur

only once or twice.

That would result in many bins of one or two

and make the histogram redundant.

If we adjust the bin size to three,

we will see that the graph slightly resembles

a normal distribution.

However, we will also notice the fatter tails.

Since the number of observations is limited

we can safely consider this sample follows

a Student's t-distribution.

Recall that the Student's t-distribution is also symmetric.

So we are confident

that even the small sample we are examining

confirms our goal of a balanced game.

Before we move on to other aspects

of the development of the game,

let's explore how a single stat is distributed

among the players in the game.

Take the shot power column for example.

If we construct a histogram and set the bin size to one,

we will see a distribution with two peaks.

It resembles two graphs placed side by side.

A way to interpret this

is having two distinct groups of players,

one with a mean of around 21

and another one with a mean of around 65.

The reason behind this

is the presence of goalkeepers in the game.

The stats important for them are completely different

from the stats essential for outfield players.

Thus, it only makes sense

that they will have distinctly lower values

for many of the non-goalkeeper specific stats.

If we examine a goalkeeping trait like GK diving,

we will be able to see the division into types more clearly.

We have two completely different clusters.

The low value represents how outfield players

would perform in goal...

And the higher one represents

the actual goalkeepers performance.

If we only examine the goalies

we will see the values are normally distributed once again

so the game is indeed balanced.

Great job.

Another aspect which meets the game more enjoyable

is creating a sense of realism.

For instance, the young professional soccer players

outnumber the veterans.

Here are the reasons why.

First, a significant number of promising young players

suffer bad injuries,

which significantly slow down their progress

or even halt it altogether.

Second, some are forced to retire

while others simply decide to quit

after spending too much time off the field.

Last but not least,

young players who are not given the opportunity to play

often decide to go to university

instead of pursuing a career in soccer.

All of these factors lead to attrition

which results in having fewer players above the age of 35

than players below the age of 20.

To make sure the game captures this aspect of the sport

check out the age column.

Once again, we can construct a histogram

and set the bin size to one.

We already demonstrated how to do this

for the overall column, so just follow the same steps.

By setting the bin width to one

every age gets represented by a separate bar on the graph.

Age is a discreet variable

representing the age of each player.

In addition, age has a minimum value of 16

since the game only consists of first team players

who have signed a professional contract.

Thus, you can consider 16 as the starting point

for any player who can sign a professional contract.

You may view it as sort of an origin

for a Poisson distribution.

Then each bar in the graph would showcase the likelihood

of a certain player within the data to be a specific age.

Since a Poisson distribution is skewed,

the younger players outnumber the older ones.

As we mentioned before, that is also true in real life.

Therefore, this creates an additional layer of realism

to the game, and should make it more enjoyable

for the customers.

Do you remember that as a head project manager,

apart from the development of the game,

you also need to supervise the official release?

One of its most important aspects is social media marketing.

Now, imagine your main competitor

is trying to expand their customer base

by uploading free video previews

of their new games each Monday prior to their launch.

A month ago, you assigned one of the interns

to keep track of the progress of their views.

You can find the recorded viewership values

in the Daily Views Excel file accompanying this lecture.

Before we proceed with the analysis

I recommend that you download and open the file.

Okay, the Excel file contains a single sheet titled Views,

which comprises two columns.

The first one indicates the number of days post-release

when the value was recorded.

The second one shows the number of views

since the last check.

To get a better understanding of the data

you would wanna see how viewership changes over time.

In order to do so, you decide to graph the data set.

The easiest way to do this

is by marking columns A and B and clicking on insert.

The next step is going to charts

and selecting a scatter plot.

Since most of the views occur within the first few days

the graph starts off at a very high point

and drops down rather quickly.

We can see that daily views start around 100,000

but fall to about 20,000 within a week.

Once the new video is released and promoted

viewership drops to around 10,000 per day

and steadily decreases as it loses relevancy.

By the time a second video has been released

around the 14th day,

the video gets barely a few thousand views per day.

This kind of behavior resembles an exponential distribution.

To check how accurate our assumption is,

we can select the chart elements button on

and select a trend line.

If we do not specify the type of relationship we expect,

Excel is going to assume a linear one

and create a straight trend line.

Since this distribution resembles an exponential one

we pick an exponential trend line instead.

The curve of the trend line fits the data points accurately.

If we assume that the views in fact follow

such a distribution,

then the trend line would represent the PDF

for a view occurring on a specific day.

To test whether views really follow

an exponential distribution,

we should look at the CDF graph as well.

We can graph the relationship between the first

and third columns.

Since total views represents the cumulative number of views

up to a given period in time,

it shows the aggregated number of views the video got.

Let's create another scatter plot

following the same steps as last time.

We can notice that the curve goes up at a decreasing rate

before eventually plateauing.

This also matches our expectation of the CDF

of an exponential distribution.

Now that we know the viewership fluctuates each day

we can state that each video loses relevancy rather quickly.

This means that such a campaign is only beneficial

in the short term.

Therefore, you advise your marketing team

to release similar videos only during the last month

before launching the game.

That way, all the videos will generate enough attention

to make the game feel immense

without losing customer interest in the process.

Fantastic.

In addition to competitor analysis

you need to conduct some customer analysis as well.

You certainly care which of your clients can afford

to spend more on in-game purchases,

so you send out a survey.

One of the survey questions

is whether the customer is a premium member

of the official fan club of any team in the game.

Since these fans are more devoted and financially capable

you wanna find out if there is any other feature

you could use to target this group.

You decide to examine a small sample of the data

which contains the age of the customer

according to their EA sports account

and whether they are a premium member.

This data is stored in the Customer's Membership Excel file

accompanying this lecture.

After opening the file, we see two columns,

one with numeric values

and the other one with ones and zeros.

The first column represents the age of the customer.

The second one shows whether they are a member or not.

If the customer is also a member of the fan club

we put one in the second column.

Alternatively, if they are not, we write down a zero.

Now, if we construct the scatter plot

we are going to see that most people under the age of 34

don't have a membership.

Whilst most people over the age of 34 do.

Of course, there are exceptions to this rule,

which is normal when we are dealing with real world data.

That being said, the data looks

like it follows a logistic distribution

since the likelihood of having a membership sharply rises

after nearing a specific value.

In this case, we can think about 34

as the location of the distribution.

This leads us to believe that 34 is the approximate age

at which customers have already reached financial stability

and can afford higher membership fees.

This insight suggests we should target customers

above the age of 34 since they're more likely to spend more.

One way to use this information

is to release more expensive legend FIFA ultimate team cards

for players who have retired in the past 20 years.

Fantastic work.

In this lecture, you were able to see numerous examples

where knowing how to deal with distributions

is truly beneficial.

You developed an understanding

of the practical aspect of probability,

and discovered why knowing how the data is distributed

can help us make correct business decisions.

In the next section of the course

we will further talk about how probability ties

into other important fields, like finance and data science.

See you all there and thanks for watching.