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.
Others, like recording time and distance in
track & field, have infinitely many outcomes.
They follow continuous distributions.
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 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This is superseded by a capital letter depicting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 dataset in parenthesis.
The characteristics are usually, mean and
variance but they may vary depending on the
type of the distribution.
Alright!
Let us start by talking about the discrete
ones.
We will get 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 discrete 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 7 native students and
3 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 3 times and trying to
calculate the likelihood of getting heads
twice.
Lastly, we should mention the Poisson Distribution.
We use 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 averages 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 adjusted 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
resemble 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 350kg and
700kg.
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 more values extremely
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 0 on the left.
Depending on the average and maximum values
within the set, the curve of the Chi Squared
graph is usually skewed to the left.
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 generates 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 cut-off point for
a successful outcome.
For instance, take a competitive e-sport like
Dota 2.
We can use a 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.
