Original subtitles

Instructor: Welcome back!

This lecture is going to serve

as an overview of what a probability distribution is

and what some of its main characteristics are.

Simply put, a distribution shows the possible values

a variable can take and how frequently they occur.

Before we start,

let us introduce some important notation we use

for the remainder of the course.

Assume that uppercase Y represents the actual outcome

of an event and lowercase Y represents one

of the possible outcomes.

One way to denote the likelihood

of reaching a particular outcome Y,

is P of Y equals Y.

We can also express it as P of Y.

For example, uppercase Y could represent the number

of red marbles we draw out of a bag

and lowercase Y would be a specific number

like three or five.

Then we express the probability

of getting exactly five red marbles

as P of Y equals five or P of five.

Since P of Y expresses the probability

for each distinct outcome,

we call this the probability function.

Good job folks!

So probability distributions

or simply probabilities measure the likelihood

of an outcome depending

on how often it is featured in the sample space.

Recall that we constructed the probability

frequency distribution of an event

in the introductory section of the course.

We recorded the frequency

for each unique value and divided it

by the total number of elements in the sample space.

Usually, that is the way we construct these probabilities

when we have a finite number of possible outcomes.

If we had an infinite number of possibilities

then recording the frequency for each one becomes impossible

because there are infinitely many of them.

For instance, imagine you are a data scientist

and want to analyze the time it takes for your code to run.

Any single compilation could take anywhere

from a few milliseconds to several days.

Often, the result will be between a few milliseconds

and a few minutes.

If we record time in seconds, we lose precision

which is something to be avoided.

To do so we need

to use the smallest possible measurement of time.

Since every milli, micro or even nanosecond could be split

in half for greater accuracy no such thing exists.

In less than an hour from now

we will talk in more detail about continuous distributions

and how to deal with them.

Now is the time to introduce some key definitions.

Regardless of whether we have a finite

or infinite number of possibilities,

we define distributions using

only two characteristics,

mean and variance.

Simply put, the mean

of the distribution is its average value.

Variance, on the other hand

is essentially how spread out the data is.

We measure this spread by how far away

from the mean all the values are.

The more dispersed the data is

the higher its variance will be.

We denote the mean of a distribution

with the Greek letter mu

and it's variance with sigma-squared.

Okay, when analyzing distributions

it is important to understand what kind

of data we are dealing with, population or sample data.

Population data is the formal way of referring

to all the data while sample data is just a part of it.

For example, if an employer surveys an entire department

about how they travel to work

the data would represent the population

of the department.

However, this same data would also just be a sample

of the employees in the whole company.

Something to remember when using sample data is

that we adopt different notations

for the mean and variance.

We denote sample mean as x-bar

and sample variance as s-squared.

One flaw of variance is

that it is measured in squared units.

For example, if you are measuring time and seconds,

the variance would be measured in seconds-squared.

Usually, there is no direct interpretation of that value.

To make further sense of variance, we introduce

a third characteristic of the distribution

called standard deviation.

Standard deviation is simply the positive square root

of variance.

As you may suspect, we denote it as sigma when dealing

with a population and as S when dealing with a sample.

Unlike variance, standard deviation is measured

in the same units as the mean.

Thus, we can directly interpret it and is often preferable.

One idea, which we will use a lot, is that any value

between mu minus sigma and mu plus sigma falls

within one standard deviation away from the mean.

The more congested the middle of the distribution,

the more data falls within that interval.

Similarly, the less data that falls

within the interval the more dispersed the data is.

Fantastic!

It is important to know that a constant relationship exists

between mean and variance for any distribution.

By definition, the variance equals the expected value

of the squared difference from the mean for any value.

We denote this as sigma-squared equals the expected value

of Y minus mu-squared.

After some simplification, this is equal

to the expected value of y-squared minus mu-squared.

As you will see in the coming lectures

if we are dealing with a specific distribution

we can find a much more precise formula.

Okay, when we are getting acquainted

with a certain data set

we want to analyze or make predictions with,

we are most interested

in the mean, variance and type of the distribution.

In our next video, we will introduce several distributions

and the characteristics they possess.

Thanks for watching!