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Hello and welcome to the next lecture in our
course Introduction to Data Analytics.
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In this lecture we are going to be talking
about Random Variables and Probability Distributions
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and this would be the first lecture of the
series that cover this topic.
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Just a recap on what we have completed so
far, we finished looking at Descriptive Statistics
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and the use of various use of various graphical
and visualization techniques in descriptive
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statistics as well as the use of summary statistics.
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Within summary statistics, we looked at measures
of centrality and measures of dispersion.
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So, jumping into this topic, quick question
is why do we need to talk about probability
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distributions.
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What does it have to do with data?
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It is just like a mathematical concept, why,
what does it have to do with data.
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And the quick answer to that question is,
if you go back to the use of the histogram
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we express that as a way of describing data.
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Essentially the histogram is, if you look
at this picture on the slide, the histogram
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is those vertical gray, grayish blue bars
that you see on this graph and that describes
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the data that summarizes the data in some
way.
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But, you might be of the belief that if you
redo this exercise, if you collect a new sample
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you will get bars that looks slightly different
and the question is, is it really coming from
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a probability distribution, is it coming from
some other mathematical function that closely
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approximates this histogram that you are seeing.
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And that red line that you see on this is
the attempt to fit this mathematical function
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and the core idea here is that, this data
is being generated by this probability distribution
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function, which is that red line and the histogram
is, what you see in terms of data.
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Because, not every time you going to get data
that looks exactly like the red line, so it
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is in this context that you can think of a
probability distribution also as a way of
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just describing your data.
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But, I would say describing and not summarizing,
because it is fairly comprehensive, it just
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does not give you one number or one thing.
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It gives the full form and shape of that data
and you can think of it as an exercise also
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in modeling your data.
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So, you are not just describing it, you modeling
it.
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So, in that context probability distributions
are very important and we will also see how
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in various other things, not just describing
data, but even in terms of doing more advanced
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analytics in the machine learning parts in
the statistical inference parts, the use of
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probability distributions is critical.
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The think of a data set has random numbers
that are being generated in accordance to
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some mathematical function is the whole idea
behind, the use of probability distributions
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with respect to data.
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To do this, we kind of have to understand
some basic concepts, which is and the first
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basic bond is to understand what random variables
are.
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Random variable is essentially a variable
whose value is subject to variations due to
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randomness as a post to variations due to
some other phenomena.
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So, we are all familiar with the concept of
constant, which just means it is a fixed number
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and variable.
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Here I am talking about the variable that
you probably learnt in algebra in high school,
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non random variables and there you learnt
that the variable is essentially something
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that can take on many possible numbers or
any possible number.
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But, the distinguishing factor between a variable
and a random variable is that, with a regular
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variable once you fix all the externalities,
then the variable takes on a specific value.
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So, let me give you an example.
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So, you take something simple like, force
is equal to mass times acceleration.
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So, force is equal to mass into acceleration
and you might say, all these are variables
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and they are and that is true.
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So, force can be any possible value, given
what I have just told you, force can take
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on various numbers as it is value with some
units.
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But, once you fix mass and you fix acceleration,
force will take on a very specific value.
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As opposed to a random variable, where even
if you fix all the externalities, the best
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way to describe the random variable would
be to say that it can still take on a set
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of possible values and that set could be a
very large set, it could be infinitely large
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set.
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But, the variable itself can take on many
possible values and each of those values have
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a specific probability associated with it
and beyond that, you are not going to, you
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cannot reduce the variable beyond that by
definition.
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Even after you fixed everything around this
variable, you still have to describe the variable
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with a probability state space.
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So, let us, the best way to again get this
even deeper, to understand this even better
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is to talk about somewhere you specific probability
distributions and that is what we are going
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to do in the next part of this class.
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But, before I proceed I just wanted to tell
you that, I am broadly breaking up the idea
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of probability distributions into discrete
probability distributions and continuous distributions
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and that is just to give you some structure
into it.
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Those words might not make immediate sense
to you right away, but what we are going to
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do right now is to look at discrete probability
distributions.
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You might also notice that I am using the
word probability density functions and you
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might not know over that word means yet, but
very soon we are going to be talking about
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that is well.
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So, great, so we are going to look at the
most simplest discrete probability distribution
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and this, it is really simple, because I think
we all used it in some sense in our daily
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life.
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So, we are here, so let us look at the first
example, I will give these numbers just show
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that going forward, it is clear.
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So, we are looking at number 1.
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In number 1, matches the closest with a colloquial
use of probability, chance, likelihood and
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so on and we are saying something simple,
which is that the probability something happens
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is x.
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So, you might say the probability that rains
today is 10 percent, is the 10 percent chance
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it is going to rain today.
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What goes unsaid is that, there is therefore,
a 90 percent chance that it does not rain
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today and that is what is captured in this
graph.
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So, we more used to saying, this is 30 percent
chance there it is going to rain, this is
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20 percent chance that there will be an accident,
there is a 10 percent chance that the product
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that I am manufacturing is not fit to be shift.
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But, essentially we are talking about these
kind of binary events, where one of the possible
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outcomes is x and therefore, by definition
the remaining possible is just 1 minus x or
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if you thinking of it in terms of percentage,
this is the 100 minus x.
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Again a very simple example of this could
also be something like this, there is a 50
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percent chance that, if I toss this coin I
am going to get a heads and what goes without
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saying is therefore, that there is a 50 percent
chance that you would not get heads.
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In this case, that is called tails, so great.
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So, that is one very simple conception of
probability and this is a probability distribution,
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it is called Bernoulli distribution, but we
can move to multiple outcomes.
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So, if you look at number 2, what we have
there is, what you get when you role a dice.
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So, you role a dice and a dice has six faces
and on each face you have a dot.
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So, you have, if you role a dice you either
get a 1 or a 2 or a 3 or a 4 or a 5 or a 6
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and the idea is that the probability of each
of these is one sixth, if it is a fair dice
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and so, that is a different kind of a probability
distribution.
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We will soon learn in our next class that
is called discrete uniformed distribution,
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because they are all the same probability,
but the possible outcomes going back to our
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definitions is set.
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So, the possible outcomes possible values
are 1, 2, 3, 4, 5, 6; the probability associated
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with each of those possible values is one
sixth and one thing that you might have noticed
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by now is, if you take all the possible values
and you take each probability and you add
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the mole up, you always get 1.
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So, in the case of 1, we saw that if the probability
of it raining was let us say 30 percent or
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let us say the probability it rains today
is 30 percent and the probability it does
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not rain therefore, becomes 70 percent.
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You add those two up you get a 100 percent
or you get 1.
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Similarly, you have six possible outcomes
when you role a dice and there is a one sixth
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chance of each of them happening and six times
1 by 6 is 1 and the intuition for this should
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also be obvious, that if you role a dice or
if the day passes means something has to have
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happened.
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So, you have had to have gotten one of those
six numbers or you know it either rained or
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it did not rain, but as long as you have comprehensively
covered the universe of possibilities, then
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something needs to have definitely happened
within that universe.
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So, therefore, that should also been intuition
us to why, that the probability distribution
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sum to 1.
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What we have in number 3 is the idea that,
again it ties to this notion of probability
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not just being a theoretical exercise and
you might actually have some data and you
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might choose to define the probability distribution
based off of what you see in the data.
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So, if somebody came to you and said look,
I do not want you to assume that, so I wanted
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to take this coin and I wanted to describe
this random variable, which is the probability
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of getting a heads or a tails and that is
the random variable and I do not want you
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to assume, there is a 50, 50 chance.
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So, you might say fine, I have nothing I cannot
assume anything and you might toss the coin
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a few times.
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So, you do a data collection exercise, where
you toss the coin 30 times and you notice
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that you get 14 heads and you get 16 tails
and for whatever reason, if you do not want
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to assume anything about the distribution
and let us say, you also do not want to do
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any statistical inference, again a topic that
we will cover soon.
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You might just be contained and saying, I
am going to describe this random variable
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with the actual data that I see.
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So, I am going to actually say that a 14 out
of 30, there is a 14 by 30, because you actually
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got 14 heads when you toss the coin 30 times.
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So, I am going to actually say 14 by 30 is
the probability of getting a heads and 16
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by 30 is the probability of getting a tails.
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So, there is nothing wrong with doing something
like that, we would have to see if that was
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actually made sense do, but nevertheless if
you said I just wanted to take data and I
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wanted to describe a probability distribution
with the data that I see, then you can definitely
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define a discrete distribution in this way.
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And now, we go on to something that is a little
more complicated, which is continuous distributions.
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In the previous case, in the discrete distributions
and let me erase this, all the ink.
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So, in the discrete distributions, what made
as discrete, were that the possible outcomes
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would discrete.
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So, it was either an event or a non event,
so that is discrete, there is no half event.
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So, there is no half event, not there.
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Same way are here, you either get a 1 or a
2 or a 3 or a 4, this set of possibilities
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that is the here x axis essentially has some
countable number of possible states and so,
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you cannot get a 1.5 when you roll a dice
and similarly, you cannot get a half head,
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half tails when you toss a coin.
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So, that is essentially the concept of it
being a discrete distribution and with continuous
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distributions; however, that is not really
true.
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The idea is that the x axis are here, so it
is the same idea which is the possible outcomes
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are on the x axis, same thing that we saw
on discrete and the probabilities are on the
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y axis.
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Case of this is the same core concept of describing
the distribution, but here the x axis is not
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discrete.
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So, what is that mean?
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What it means is, you take something like
the probability of a certain height.
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A height can be a 130 centimeters, so that
could be one number out here, but it could
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also be 130.001 centimeters, it could also
be a 129.999 centimeters.
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So, there is no inherent discretization.
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You might turn around and say, look what if
I had a measuring scale that could only measure
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in 5 centimeter intervals.
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So, the way I mean or I can only measure up
to a centimeter, I cannot measure less than
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a centimeter, because the scale that I have
does not have more resolution and that is
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fine.
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You know, if your resolution for measurement
is still accurate, meaning that anything between
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130 and 131 gets called a 130, because of
the inherent resolution of the scale.
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That is fine, you can create a discretized
version of it, but the idea is this nothing
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about height or any, essentially the measurement
of space.
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There is nothing about height that is inherently
discretized, like the measurement of a dice
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is inherently discretized.
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You cannot possibly roll the dice and get
a two and half, whereas if you had a five
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and half measure of height, you could get
any possible value within the certain range.
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So, you might have the lower end of this being
20 centimeters and the upper end of this being
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a 180 centimeters.
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But, essentially any value between it is possible
and we kind of spoke about the same concept
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when we spoke about discrete and continuous
variables.
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The same concept of discrete and continuous
variables applies to discrete and continuous
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distributions.
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Now, again another important thing to note
is just like in the discrete distributions,
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where we said the sum of all the probabilities
for each possibility should add up to a 100
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percent or should add up to 1.
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Here you cannot have a countable number of
possibilities, so you cannot take each possibility.
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Take the probability of that and added to
1, just because there are infinite number
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of them.
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So, you basically what you do is, you take
the entire interval and sum the probability
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within that interval and the best way to do
that is to look at this area as a whole.
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So, this way when you look at this area, it
is like you are taking all the possibilities
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within that area and summing all the probabilities
for each possibility and when you do that,
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you will be getting 1 or a 100 percent.
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So, now let us now that we understand both
continuous and discrete random variables.
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Letâ€™s just briefly talk about the use of
probability density functions and cumulative
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density functions.
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So, far what I have been graphically showing
you, have all bend probability density functions.
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For each probability density function, there
exist a cumulative density function and so,
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I will describe it in the continuous case
and that is the easiest and I separately do
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that for the discrete.
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The idea with the PDF is that, like we said
for, because there are infinite number of
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possible states, you really does not really
make sense to ask the question, what is the
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probability at a given point.
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It turns out that the answer to that question
is that the probability of that given point
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is zero, although there is some y, there is
some height for that given point.
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Because, there are infinite such points out
here, because there are technically infinite
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such points.
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As a result, you can only say what is the
probability of a given area, so you can say
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I want to look at this area and you can get
the answer to that question, you can get the
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probability of this given area by just measuring
the area under this curve and that is what
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I have done with the gray line on this graph
as well.
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But, the idea behind PDFs to CDF is that,
the CDF describes the cumulative probability
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up to a certain point.
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So, if I would ask the question, what is the
probability within this area, you could answer
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it from me by looking at the area under the
curve.
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If I ask the probability density function
at this given point, you can use the function
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to figure out what this value is, but the
probability itself is zero.
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But, the cumulative describes the probability
from zero, from the lower end of this axis
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and that can be zero or that could be something
like minus infinity or it could be some other
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value.
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You know this starting point essentially,
from that starting point all the way up to
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the point of interest.
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So, this is x, the PDF describes the height
out here for x, the CDF describesâ€¦
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So, this is your point x, the PDF describes
the height of this curve at x and which is
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y, this entire curve out here this curve that
is here on the left hand side is the PDF,
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whereas the CDF describes the area to the
left of this point x and that area is the
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CDF.
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00:20:49,440 --> 00:20:57,919
So, this graph is nothing but, the same as
the graph to the left, where at each point
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x you are looking at what is the area to the
left of x on the PDF and that is what you
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are plotting here and that should logically
be equal to 1, when you complete and the reason
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for it is the following.
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We already discussed in the previous section
is to how the overall area under this curve.
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The overall area under this curve is equal
to 1, correct.
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We discussed that if you take all the possible
states and add the probabilities of all the
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possible states, as to how you are getting
up, you would get a probability of a 100 percent
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or 1 and.
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So, the CDF is nothing but, this description
of the area to the left of the curve and when
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00:21:44,850 --> 00:21:52,700
you reached your entire set of possible heights
or in this particular case heights, but it
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can be anything else, then the CDF hits at
the one mark when it ends.
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And it is, it would be helpful for you to
know that therefore, the easiest way of getting
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from a PDF to CDF and a CDF to PDF is, from
a PDF to CDF you would essentially want to
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integrate.
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For those of you, who used integration in
high school or if you heard of the term integration,
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that is that symbol that looks like this and
the idea is that an integration covers this
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core concept of area under the curve.
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00:22:30,890 --> 00:22:40,399
So, if you integrate up to a point x, so let
us say just for as an example that this started
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at zero, but you could change that.
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00:22:42,770 --> 00:22:47,880
It could start y, it could started minus infinity,
but you essentially if you integrate the area
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under this curve to get the CDF.
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You would just want to integrate from 0 to
x, some f of x and the f of x here is your
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PDF function and that will give you the CDF
and the idea of going from CDF to PDF is exactly
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the opposite, which is to differentiate this
CDF and that will give you the PDF.
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So, we will conclude this lecture with that
note and starting from the next lecture, we
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will be talking about some actual probability
distributions and we will go through at least
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the most popular ones in that lecture.
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Thank you.