Original subtitles

-: Hello again.

In this lecture, we are going to talk about

various types of Probability distributions

and what kind of events they can be used to describe.

Certain distributions share features,

so we group them into types.

Some like rolling a die or picking a card,

have a finite number of outcomes.

They follow Discrete distributions,

and we use the formulas we already introduced

to calculate their probabilities and expected values.

Others like, recording time and distance in track and field,

have infinitely many outcomes.

They follow Continuous distributions.

And we use different formulas

from the ones we mentioned so far.

Throughout the course of this video,

we are going to examine the characteristics

of some of the most common distributions.

For each one, we will focus on an important aspect of it

or when it is used.

Before we get into the specifics,

you will need to know the proper notation we implement

when defining distributions.

We start off by writing down the variable name

for our set of values,

followed by the Tilde sign.

This is superseded by a capital letter

depicting the type of the distribution

and some characteristics of the data set in parenthesis.

The characteristics are usually mean and variance,

but they may vary depending on the type of the distribution.

All right, let us start by talking about the discrete ones.

We will give an overview of them

and then we will devote a separate lecture to each one.

So we looked at problems relating to drawing cards

from a deck or flipping a coin.

Both examples show events where all outcomes

are equally likely.

Such outcomes are called equiprobable,

and these sorts of events follow a uniform distribution.

Then there are events with only two possible outcomes,

true or false.

They follow a Bernoulli distribution.

Regardless of whether one outcome is more likely to occur,

any event with two outcomes can be transformed

into a Bernoulli event.

We simply assign one of them to be true

and the other one to be false.

Imagine we are required to elect a captain

for our college sports team.

The team consists of seven native students

and three international students.

We assign the captain being domestic to be true,

and the captain being an international as false.

Since the outcome can now only be true or false,

we have a Bernoulli distribution.

Now, if we carry out a similar experiment

several times in a row,

we are dealing with a Binomial distribution.

Just like the Bernoulli distribution,

the outcomes for each iteration are two,

but we have many iterations.

For example, we could be flipping the coin

we mentioned earlier three times,

and trying to calculate the likelihood

of getting heads twice.

Lastly, we should mention the Poisson distribution.

We used it when we want to test out

how unusual an event frequency is for a given interval.

For example, imagine we know that so far

LeBron James has averaged 35 points per game

during the regular season.

We want to know how likely it is

that he will score 12 points in the first quarter

of his next game.

Since the frequency changes,

so should our expectations for the outcome.

Using the Poisson distribution,

we are able to determine the chance of LeBron

scoring exactly 12 points for the specified time interval.

Great.

Now on to the Continuous distributions.

One thing to remember,

is that since we are dealing with continuous outcomes,

the Probability distribution would be a curve

as opposed to unconnected individual bars.

The first one we will talk about is the Normal distribution.

The outcomes of many events in nature

closely resembled this distribution, hence the name normal.

For instance, according to numerous reports

throughout the last few decades,

the weight of an adult male polar bear

is usually around 500 kilograms.

However, there have been records of individual species

weighing anywhere between 350 kilograms and 700 kilograms.

Extreme values like 350 and 700 are called outliers

and do not feature very frequently in Normal distributions.

Sometimes we have limited data for events

that resemble a normal distribution.

In those cases, we observe the Student's-T distribution.

It serves as a small sample approximation

of a Normal distribution.

Another difference is that

the Student's-T accommodates extreme values

significantly better.

Graphically, that is represented by the curve

having fatter tails.

Overall, this results in a larger number of values

located far away from the mean.

So the curve would probably more closely resemble

a Student's-T distribution than a Normal distribution.

Now, imagine only looking at the recorded weights

of the last 10 sightings across Alaska and Canada.

The lower number of elements would make the occurrence

of any extreme value

represent a much bigger part

of the population than it should.

Good job everyone.

Another Continuous distribution

we would like to introduce,

is the Chi-Squared distribution.

It is the first asymmetric

Continuous distribution we are dealing with

as it only consists of non-negative values.

Graphically, that means that the Chi-Squared distribution

always starts from zero on the left.

Depending on the average and maximum values within the set,

the curve of the Chi-Squared graph

is typically skewed to the right.

Unlike the previous two distributions,

the Chi-Squared does not often mirror real-life events.

However, it is often used in hypothesis testing

to help determine goodness of fit.

The next distribution on our list

is the Exponential distribution.

The Exponential distribution is usually present

when we are dealing with events

that are rapidly changing early on.

An easy to understand example

is how online news articles generate hits.

They get most of their clicks when the topic is still fresh.

The more time passes,

the more irrelevant it becomes and interest dies off.

The last Continuous distribution we will mention

is the Logistic distribution.

We often find it useful in forecast analysis

when we try to determine a cutoff point

for a successful outcome.

For instance, take a competitive Esport like Dota 2.

We can use the Logistic distribution

to determine how much of an in-game advantage

at the 10-minute mark is necessary

to confidently predict victory for either team.

Just like with other types of forecasting,

our predictions would never reach true certainty,

but more on that later.

Whoa, good job, folks.

In the next video,

we are going to focus on Discrete distributions.

We will introduce formulas for competing expected values

and standard deviations

before looking into each distribution individually.

Thanks for watching.