Random Variables and Probability Distributions Video Lecture | What is Data Analytics? - Data & Analytics

hello and welcome to the next lecture in
our course introduction to data
analytics in this lecture we are going
to be talking about random variables and
probability distributions and this will
be the first lecture of a series that
cover cover the stop
just to recap on what we have completed
so far we finished looking at
descriptive statistics and the use of
various views of graphical and
visualization techniques in descriptive
statistics as well as the use of summary
statistics within summary statistics we
looked at measures of centrality and
measures of dispersion so jumping into
this topic quick question is why do we
need to talk about probability
distributions what does it have to do
with data is it just like a mathematical
concept why what does it have to do with
data and the quick answer to that
question is if you go back to the use of
the histogram be expressed that as a way
of describing data essentially the
histogram is if you look at this picture
on the slide the histogram is those
vertical gray grayish blue bars that you
see on this graph and that describes the
data that summarizes the data in some
way but you might be of the belief that
if you redo this exercise if you collect
a new sample you'll get bars that look
slightly different and the question is
is it really coming from a probability
distribution is it coming from some
other mathematical function that closely
approximates this histogram that you're
seeing and that red line that you see in
this is the attempt to fit this
mathematical function and the core idea
here is that this data is being
generated by this probability
distribution function which is that red
line and the histogram is what you see
in terms of data because not every time
are you going to get data that looks
exactly like the red line so it's in
this context that you can think of a
probability distribution also as a way
of just describing your data but it's it
I wouldn't i would say describing and
not summarizing because it's fairly
comprehensive it just does not give you
one number or one thing it gives you the
full form and shape
of the data and you can think of it as
an exercise also in modeling your data
so you so you're not just describing it
you're modeling it so in that context
probability distributions are very
important and we will also see how in
various other things not just describing
data but even in terms of doing more
advanced analytics in the machine
learning parts in the statistical
inference parts the use of probability
distributions is critical to think of a
data set as random numbers that are
being generated in accordance to some
mathematical function is the whole idea
behind the use of probability
distributions with respect to data okay
to do this we kind of have to understand
some basic concepts which is and the
first basic one is to understand what
random variables are a random variable
is essentially a variable whose value is
subject to variations due to randomness
as opposed to variations due to some
other phenomena right so we're all
familiar with the concept of a constant
which just means it's a fixed number and
variable here I'm talking about the
variable that you probably learned in
algebra in high school
non random variables and there you learn
that a variable is essentially something
that can take on many possible numbers
or any possible number but the
distinguishing factor between a variable
and a random variable is that with a
regular variable once you fix all the
externalities then the variable takes on
a specific value so let me give you an
example so you take something simple
like force is equal to mass
times acceleration so force is equal to
mass into acceleration and you might say
yeah all of these are variables right
and they are and that's true so force
can be any possible value given what
I've just told you force can take on
various numbers as its value with some
units but once you fix mass and you fix
acceleration force will take on a very
specific value as opposed to a random
variable where
even if you fix all the externalities
the best way to describe the random
variable would be to say that it can
still take on a set of possible values
and that set could be a very large set
it could be an infinitely large set but
the variable itself can take on many
possible values and each of those values
have a specific probability associated
with it and beyond that you are not
going to dis you you cannot reduce the
variable beyond that by definition even
after you fixed everything around this
variable you still have to describe the
variable width with a probability state
space so let's the best way to again get
this even deeper to understand this even
better is to talk about some very
specific probability distributions and
that's what we're going to do in the
next part of this class but before I
proceed I just wanted to tell you that
I'm broadly breaking up the the idea of
probability distributions into discrete
probability distributions in continuous
distributions and that is just to give
you some structure you those words might
not make immediate sense to you right
right away but what we're going to do
right now is to look at discrete
probability distributions you might also
notice that I'm using the word
probability density functions and you
might not know what that word means yet
but very soon we're going to be talking
about that as well so great so we're
going to look at the most simplest
discrete probability distribution and
this it's really simple because I think
we've all used it in some sense in our
daily lives so we're here so let's look
at the first example I'll give these
numbers just so that going forward it's
clear okay so we're looking at number
one number one matches the closest with
a colloquial use of probability chance
likelihood and so on and we're saying
something simple which is that the
probability is something have
is X so you might say the probability
that vanes today is 10% there's a 10%
chance it's going to rain today what
goes unsaid is that there is therefore a
90% chance that it does not rain today
and that's what is captured in this
graph so we're more used to saying
there's a 30% chance that it's going to
rain there's a 20% chance that there
will be an accident
there is a 10% chance that the product
that I'm manufacturing is not fit to be
shipped but essentially we are talking
about these kind of binary events where
one of the possible outcomes is X and
therefore by definition the remaining
possible is just 1 minus X or if you're
thinking of it in terms of percentages
it's 100 minus X again a very simple
example of this could also be something
like there's a 50% chance that if I toss
this coin I'm going to get a heads and
what goes without saying is therefore
that there is a 50% chance that you will
not get hits in this case that's called
tails so great so that's that's one very
simple conception of probability and
this is a probability distribution it's
called a Bernoulli distribution but we
can move to multiple outcomes right so
if you look at number two what we have
there is what you get when you roll a
dice so you roll a dice and a dice has
six faces and on each face you have dots
so you have if you roll a dice you
either get a 1 or 2 or 3 or 4 or 5 or 6
and the idea is that the probability of
each of these is 1/6 if it is a fair
dice and so that's a different kind of a
probability distribution we will soon
learn in our next class that that's
called a discrete uniform distribution
because they're all the same probability
but the possible outcomes going back to
our definitions is that so the possible
out
possible values are one two three four
five six the probability associated with
each of those possible values is 1/6 and
one thing that you might have noticed by
now is if you take all the possible
values and you take each probability and
you add them all up you always get 1
right so in the case of 1 we saw that if
the probability of it's raining was
let's say 30% let's say the probability
rains today is 30% and the probability
does not rain therefore become 70% you
add those two up you get 100% or you get
1 similarly you have six possible
outcomes when you roll a dice and
there's a 1/6 chance of each of them
happening and 6 times 1 by 6 is 1 and
the intuition for this should also be
obvious that if you roll a dice or if
the day passes I mean something has to
have happened right so you have had to
have gotten one of those six numbers or
you know it either rained or did not
rain but as long as you have
comprehensively covered the universe of
possibilities then something needs to
have definitely happened within that
universe so therefore that should also
be an intuition as to why that the
probability distribution sum to 1 what
we have in number 3 is the idea that
again it ties to this notion of
probability not just being a theoretical
exercise and you might actually have
some data and you might choose to define
the probability distribution based off
of what you see in the data so if
somebody came to you and said look I
don't want you to assume that so I want
you to take this coin and I want you to
describe this random variable which is
the probability of getting a heads or a
tails and that's the random variable and
I don't want you to assume there's a
50-50 chance
so you might say fine I have nothing I
cannot assume anything and you might
toss the coin a few times so you do it a
data collection exercise where you toss
the coin 30 times and you notice that
you get 14 heads and you get 16 tails
and for whatever reason if you don't
want to assume anything about the
distribution and let's say you also
don't want to do any statistical
inference again a topic that we will
cover some you might just be content in
saying I'm going to describe this random
variable with the actual data that I see
so I'm going to actually say that a 14
or there's a 14 by 30 because you
actually got 14 heads when you toss the
coin 30 times so I'm going to actually
say 14 by 30 is the probability of
getting a heads and 16 by 30 is the
probability of getting tails so there's
nothing wrong with doing something like
that we we would have to see if that
that was that actually made sense to do
but nevertheless if you said I just want
to take data and I want to describe a
probability distribution with the data
that I see then you can definitely
define a discrete distribution in this
way now we go on to something that's a
little more complicated which is
continuous distributions in the previous
case in the discrete distributions and
let me erase this all linked so in the
discrete distributions
what made it discrete were that the
possible outcomes were discrete so it
was either an event or a non-event so
that's that's discrete there's no half
event right so there's no half event not
there same way out here you either get a
1 or a 2 or a 3 or a 4 this set of
possibilities that is the ear x-axis
essentially has some countable number of
possible States and so you cannot get a
1.5 when you roll a dice and similarly
you can't get a half head half tails
when you toss a coin so that's
essentially the concept of it being a
discrete distribution
and with continuous distributions
however that's not really true the idea
is that the x-axis out here so it's the
same idea which is the possible outcomes
or on the x-axis same thing that we saw
discrete and the probabilities are on
the y-axis okay so this is the same core
concept of describing a distribution but
here the x-axis is not discrete so what
does that mean what it means is you take
something like the probability of a
certain high height can be a hundred and
thirty centimeters so that could be one
number out here but it could also be a
hundred and thirty point zero zero one
centimeters it could also be one hundred
and twenty nine point nine nine nine
centimeters so there's no there's no
there's no inherent discretization you
might turn around and say look what if I
had a measuring scale that could only
measure in five centimeter intervals so
the way I mean or I can only measure up
to a centimeter I can't measure less
than a centimeter because the scale that
I have does not have more resolution and
that's fine you know if you know if your
resolution for measurement is still
accurate meaning that anything between
130 and 131 gets called 130 because of
the inherent resolution of the scale
that's fine you can create a discretized
version of it but the idea is there's
nothing about height or the essentially
the measurement of space
there's nothing about height that is
inherently discretized like the
measurement of a dice right is
inherently discretized you cannot
possibly roll the dice and get a two and
a half whereas if you had
a fine enough measure of height you
could get any possible value within a
certain range so you might have the
lower end of this being 20 centimeters
and the upper end of this being 180
centimeters but essentially any value
between it is possible and we kind of
spoke about the same concept when we
spoke about discrete and continuous
variables the same concept of discrete
and continuous variables applies to
discrete and continuous distributions
now again another important thing to
note is just like in the discrete
distributions where we said the sum of
all the probabilities for each
possibility should add up to a hundred
percent or should add up to one here you
cannot have a countable number of
possibility so you can't take each
possibility take the probability of that
and add it to one just because there are
infinite number of them so you basically
what you do is you take the entire
interval and sum the probability within
that interval and the best way to do
that is to look at this area as a whole
right so this way when you look at this
area it's like you're taking all the
possibilities within that area and
summing all the probabilities for each
possibility and when you do that you
will be getting one or a hundred percent
okay so now let's know that we
understand both continuous and discrete
random variables let's just briefly talk
about the use of probability density
functions and cumulative density
functions so far what I have been
graphically showing you have all been
probability density functions for each
probability density function there
exists a cumulative density function
okay and so I will describe it in the
continuous case and and that's the
easiest and I'll separately do that for
the discrete the idea with the PDF is
that like we said um at four
because there are infinite number of
possible states you really it doesn't
really make sense to ask the question
what is the probability at a given point
it turns out that the answer to that
question is that the probability to the
given point is zero although there is
some why there's some height for that
given point because there are infinite
such points out here because there are
in there are technically infinite such
points as a result you can only say what
is the probability of a given area so
you can say I want to look at this area
and you can get the answer to that
question you can get the probability of
this given area by just measuring the
area under this curve and that's what
I've done with a gray line on this graph
as well but the idea behind PDFs to CDF
is that the CDF describes the cumulative
probability up to a certain point so if
I were to ask the question what is the
probability within this area you could
answer it for me by looking at the area
under the curve if I ask you the
probability density function at this
given point you can use the function to
figure out what this value is but the
probability itself is here but the
cumulative describes the probability
from 0 from the lower end of this axis
and that can be 0 or that could be
something like minus infinity or it
could be in some other value you know
this the starting point essentially from
that starting point all the way up to
the point of interest so if this is X
right the PDF describes the height out
here for X the CDF describes ok so this
is your point X the PDF describes the
height of this curve at X and which is
why this entire curve out here this
curve that's here on the left
is the PDF whereas the CDF describes the
area to the left of this point X and
that area is the CDF so this graph is
nothing but the same as the graph to the
left where at each point X you're
looking at what is the area to the left
of X on the PDF and that's what you're
plotting here and that should logically
be equal to 1 when you complete and the
reason for it is the following we've
already discussed in the previous
section as to how the overall area under
this curve overall area under this curve
is equal to 1
correct we discussed that if you take
all the possible states and add the
probabilities of all the possible states
as to how you're getting a you get you
would get a probability of a hundred
percent or 1 and so CDF is nothing but
this description of the area to the left
of the curve and when you've reached
your entire set of possible Heights or
in this in this particular case Heights
but it can be anything else then the CDF
hits at the 1 mark now when it ends so
and it's it would be helpful for you to
know that that therefore the easiest way
of getting from a PDF to CDF in a CDF to
PDF is from a PDF to CDF you would
essentially want to integrate for those
of you who used integration in high
school or have heard of the term
integration that's that symbol that
looks like this and the idea is that an
integration covers this core concept of
area under the curve so if you integrate
up to a point x so let's say let's say
just for as an example that this started
at 0 but you could change that it could
start at Y it could start at minus
infinity but you essentially if you
integrate the area under this curve
right to integrate the area under this
curve to get the CDF you would just want
to integrate from 0 to X some f of X
right and the f of X here is your PDF
function and that will give you the CDF
and the idea of going from CDF to PDF is
exactly the opposite which is to
differentiate the CDF and that will give
you the PDF okay so we will conclude
this lecture with that note and starting
from the next lecture we will be talking
about some actual probability
distributions and we will go through at
least the most popular ones in that
lecture thank you