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Hello there and welcome to the course probability
methods in civil engineering. Today,
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we are starting a new module, module 3.
.
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In this module, we learn about random variables
and there will be couple of lectures and
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in this first lecture, we learn the concept
of random variables and their definitions.
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..
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So our outline for today’s lecture is arranged
broadly like this. First, we will discuss
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about concept of random variable and then,
we will understand the definition of random
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variable. Then we will discuss different types
of random variables and their probability
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distributions.
This random variables are important in the
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sense, that this is the, this concept is
important to understand how it is used in
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the probability theory and this probability
theory for this random variable is generally
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assessed through their distributions. So this
is known as the probability distribution of
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random variables. After that, we will learn
different properties of random variables.
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..
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So, as just what we are telling, that this
concept of this random variable, which is
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abbreviated as RV, is most important to the
probability theory and its application. Now,
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if we see that what actually this random variable
mean, if this shaded area, what you see
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here is the sample space, then this sample
space consist of all feasible outcome of one
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experiment.
Now, this feasible outcome of this experiment
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can be continuous or can be discrete
points from through this sample space. Now,
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from this sample space, we have to, for the
mathematical analysis, we have to map the
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outcome of the experiment, of one
experiment to some number, according to our
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convenience.
Now to map this each and every output of a
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random experiment to this real line, is
generally through a functional correspondence
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and this correspondence, this functional
correspondence is one random variable.
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So, what is important and what is, what we
want to understand from this slide is that,
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this
random variable, even though this variable
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is shown here in the in the name of the
random variable, is not a variable. So, which
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is generally, what you want to say is a
functional correspondence or is a function.
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That function generally map between the
sample space of one experiment to the real
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line.
Now, this mapping is done based on our convenience
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and that we will see in a minute.
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..
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Thus, if we say the definition, if you say,
then we will say that a random variable is
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a
function denoted by X or X some specific number,
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it can be denoted by any letter.
Generally, these things are denoted by this
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small lower case of this random variable.
These random variables are generally expressed
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in terms of the upper case letters. So,
this random variable X is a function that
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map each points, that is a outcome of an
experiment over a sample space to the, to
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a numerical value on the real line. What just
we have shown in this figure, in the last
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slide.
Now, before we discuss some of the, some of
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the further concept of this random variable,
so there are different notations are being
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used to denote this random variable and the
specific value. For this lecture or you will
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see in some of the classical text book, that
this
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notation will be followed for this course.
Generally, this random variable are denoted
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by
the uppercase letters and its specific values
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are denoted as lowercase letters. Thus, in
this
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notation, this uppercase letter is indicating
the random variable, which is denoted by this
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X and any specific value to this random variable
is denoted by this small x, which is
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denoted, which is shown here. So, this is
the specific value of the random variable
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X.
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..
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Now, we will see some example of this random
variable which we just discussed.
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Suppose that, number of rainy days in a month;
so, if we can, you can say that this is a
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random variable and we do not know. So, this
value that number of rainy days in a
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month can vary and if we denote this random
variable as letter capital D and some value
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say d equals to 5. So, this indicates that
5 rainy days are observed in a particular
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month.
Now, which is, which can expressed by d in
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parenthesis d equals to 5.
Second example, say the number of road accident
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over a particular stretch of the road in
a year. Suppose, this is denoted by X and
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some specific values, such as say x equals
to
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10, indicates that there are 10 road accidents
over that stretch of the road has occurred
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in
a particular year. Now, this value 10 or this
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value 5 can change in the successive time
step or in the success; for in this case,
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in the successive year, that can change. So,
this
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number of road accidents is my random variable,
which for a particular year it takes a
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specific value x equals to 10, which is denoted
by this.
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The last one, say the strength of concrete.
If, I designate it at C, so this is denoted
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by C
and some specific value such as c equals to
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25 Newton per millimetre square indicates
that 25 Newton per millimetre square strength
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was observed for a particular sample of
concrete and which is expressed by C in parenthesis
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c equals to 5. C is the, sorry this
shall be c equals 25; sorry for this mistake
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this should be 25.
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..
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So, a little bit formal definition of random
variable, if we see, that a random variable
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X is
a process to assign a number, which is a real
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number X(ζ) to every outcome ζ of a
random experiment. So, if we just take one,
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take that thing, so suppose that you are
taking that example of that of tossing a dice,
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then we know that whatever the numbers
that generally comes, if it comes a 6 on the
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top, then we see, we say that the outcome
that
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is coming, it is my 6. Similarly, six different
surfaces can have six different numbers and
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that can be happened.
.
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.Now, see that in this experiment, if I just
change this one, so this is quite straight
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forward
that, whatever number of dots that we are
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usually used to know. Suppose that, instead
of
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having this kind of dice, if I just have a
dice which is having different colours. Suppose
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that, this is having a blue colour, this side
is having my say red colour, this side is
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having
green colour and similarly six different faces
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having six different colours. Then, what I
am trying to do is that, this outcome of this
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is a top surface, the colour of the top surface,
if I just want say that this is my outcome
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of this experiment, then what I can do is
that I
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can take different colour code and I can assign
to some number, not necessarily 1 to 6, it
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can be of any number.
Say that B, I am giving some number 10, R
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giving some number 20, green giving some
number 30 and this way say up to 60 this number
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is given. So, these mapping from this
outcome to some number, this mapping is your
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random variable. So, we will come after
this, how to assign the probabilities. Now,
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this is very straight forward. So each and
every outcome, if this dice is fair, then
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the outcome of each and every of this outcome
will be equal; so everything will be 1 by
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6.
So, that is again another process. How we
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will assign the probability to that particular
random variable? If it is X, so that will
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be, how we assign the probability. Now, when
we
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are defining this random variable itself,
so this random variable is nothing but, is
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a
process to assign a number to each outcome
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of that random experiment. This is exactly
what we are calling it as a formal definition
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of this random variable is that, a random
variable X is a process to assign a number
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X(ζ) to every outcome ζ of a random
experiment. The resulting function must satisfy
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the following two conditions.
Even though, we are saying that this is a
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function, so this function of mapping from
this
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outcome, experimental outcome to some numbers
should follow two, such two
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conditions. The first condition is the set
X less than equals to small x. Now, you see
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here,
this is the capital letter which is your random
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variable and this is the specific value. Now,
this random variable, below some specific
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value is an event for every x. So, whatever
the
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x you take, so this should constitute and
event for that random variable.
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For that, for this x, now other things of
probabilities of the events, that X equals
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to plus
infinity and X equals to minus infinity are
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equal to 0. Why this things we are just saying
is that, even though means mathematically,
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this X can take any value on the real line
but
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.still, we say that so the extreme, that is
plus infinity and minus infinity, this probabilities
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if I take this probabilities, this should
be equals to 0.
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.
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A complex random variable, so just we will
to complete this definition part, if we do
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not
say this one, it will not be completed. So,
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this is, there is another type, this is the
real
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random variable which just we discussed, is
now it can be the complex random variable
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as well, which is say z equals to x plus jy.
So, this random variable is only defined by
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their joint distribution. So, if I want to
define
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the complex random variable z, this should
be defined by their joint distribution. This
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joint distribution will be discussed later
in this course. So, but for the time being,
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you can
say that this joint distribution is generally
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obtained from that distribution, of this x
and y
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both. So this both, this distribution should
be taken care to find the joint distribution
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to
define the distribution of the complex random
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variable, of this, of the random variable,
of
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individual x and y. Since, this inequality
because, so this kind of inequality, say this
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of
the random variable and this is the less than
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equal to that number does not exist. But,
for
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this course, if it is not stated specifically,
all the random variables are real. We should
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follow this norm.
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..
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Now, with this concept, we will just see different
types of random variables. There are
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basically two types of random variables; one
is the discrete random variable and other
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one is continuous random variable. But, there
is, there are some times, in particularly
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in
the civil engineering application, we have
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seen there are some distributions which can
be
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treated as the mixed random variable as well.
So, we will discuss these different types
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of
random variable one after another now. The
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first we take this discrete random variable.
If
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the possible set of values that a random variable
can define is finite, it is a discrete
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random variable.
Now, so just to tell this thing here, that
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suppose that this is my sample space. Now,
if I
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say that these random variable is defined
for some specific value and rest of the part
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it is
not defined, so it takes some specific value.
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So, those values only it can define. So, in
that case, what we say that this is a discrete
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case. So, in the previous example, if we just
see here, that it can define that some specific
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outcome of this random variable. It cannot,
if it is not that continuous, that a range
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cannot be defined here. So, if it is dots,
then we
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can say that it is defined for the outcome
1, defined of the outcome 2. But, it is not
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defined for the outcome anything any number
in between these two.
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Similarly, if we just see from this remote
sensing concept, that colour code, then we
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can
say that it is defined for the blue for this
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coloured surface, it is defined for the blue,
it is
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defined for the red but, in between mixture.
I think you know this colour code. This
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.stands for this blue, if it is the colour
are R G B are the primary colours and this
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1 0 0 is
for the red. Now, you can just change this
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number to some other to get some in between
colour. So, that is not defined there. So
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what we want the say for this discrete is
the,
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these random variables are described only
for some specific outcome of this experiment.
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It is not defined for this all. So, those
random variables whose outcome who can map,
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who can define the outcome of the random experiment
for some specific values, those
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are known as the discrete random variable.
Thus, if the possible set of values that a
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random variable can define is finite, it is
discrete
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random variable. The probabilities are defined
for the specific values which are greater
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than or equal to 0. So, this we will see in
detail which are greater than or equal to
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0. But,
what we are trying to stress here is that,
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the probabilities are defined for the specific
value. So, it is just like that, for that
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point this probabilities are defined for the
adjoining
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point, this is not defined.
Summation of all the probabilities, all the
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possible values are equals to 1. These two
things are basically coming from the axioms
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of the probability theory that we discussed
in the previous classes.
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.
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The next type is continuous random variable.
So, on the other end, in contrast to this
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discrete random variable, if the possible
set of the values that a random variable can
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define is infinite, it is continuous random
variable.
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.So over a zone, over a range this random
variable is, if it is also defined, then this
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is
known as the continuous random variable. The
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probability of any specific value is 0.
Now, this is coming from this density point
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of view. Now, once we are saying that this
probability of any specific value, if I just
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take, then this specific value probability
is 0.
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This is again coming from the axioms of this
probability. But, even though we say that
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it
is exactly, specific value it is 0 but, whatever
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small set of the values if we think, that
can
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be greater than or equal to 0. So, whatever
the small range, if we consider, then it will
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be
0. I think this will be more clear from this
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pictorial view here.
.
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So, suppose what we are trying to say that,
if there are some specific outcome and those
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probabilities are assigned like this. Whatever
may be the probabilities, if it is not these
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are like the concentrated mass for that specific
outcome.
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So, this is this is your discrete random variable.
Now, for this continuous random
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variable, what is defined is that it looks
like this. So over this zone, from this to
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this
zone, for any specific value, this is nothing
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but, the density of this one. Now following
the axioms of this probability, if we just
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adapt with this axis, is your probability.
As if, I
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adapt all this values to what we just discussed,
this will be equal to 1. But, in this case,
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when we are talking about, this is one continuous
distribution then, what happens for a
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specific point, the probability is 0 because
this is, basically, a density. Now, whatever
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the
small area that we will consider, then this
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area is giving you the probability for this
small
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.zone. Now, for the full zone, if I just take
then the total probability, that is, total,
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this is
your feasible range. So the total probability
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in this feasible range should be equals to
1.
212
00:19:19,490 --> 00:19:22,980
That comes from the axioms of this probability
theory.
213
00:19:22,980 --> 00:19:29,980
Now, again here, the next type what comes
is the mixed distribution. So mixed
214
00:19:29,980 --> 00:19:34,430
distribution, sometimes what happens for some
specific value, it is defined some
215
00:19:34,430 --> 00:19:40,679
probability. Some probabilities concentrated
here as a mass and for the rest of some
216
00:19:40,679 --> 00:19:49,750
range it can be continuous and goes up to
some level or up to the end of this or up
217
00:19:49,750 --> 00:19:54,050
to the
infinity. Now, this for this particular point,
218
00:19:54,050 --> 00:19:56,650
this probability mass is concentrated here
and
219
00:19:56,650 --> 00:19:59,470
for the rest of the region, it is having some
continuous things.
220
00:19:59,470 --> 00:20:06,560
In such cases, we say that these kind of random
variables are mixed random variable. So,
221
00:20:06,560 --> 00:20:15,820
this is the third thing, what we are seen
that. So, mixed random variable says, if for
222
00:20:15,820 --> 00:20:19,360
some
range of possible set of the values of a random
223
00:20:19,360 --> 00:20:24,280
variable is discrete and for the other range
of possible set it is continuous, it is a
224
00:20:24,280 --> 00:20:29,070
mixed random variable. Now, with these three
type
225
00:20:29,070 --> 00:20:38,630
of random variables, some example in the civil
engineering context, we will see now.
226
00:20:38,630 --> 00:20:39,630
.
227
00:20:39,630 --> 00:20:43,380
The first is that, we discuss is the discrete
random variable. Say that, this number of
228
00:20:43,380 --> 00:20:46,549
rainy
days. Now number of rainy days at a particular
229
00:20:46,549 --> 00:20:53,240
location over a period of one month say,
so it can take, either the specially the integer
230
00:20:53,240 --> 00:20:59,790
value, either 1 day or 2 days 3 days, in this
way up to 30 or 31 days, whatever may be the
231
00:20:59,790 --> 00:21:00,790
case.
232
00:21:00,790 --> 00:21:07,970
.But, it can never take any value in between
two integers. So, as this can take only some
233
00:21:07,970 --> 00:21:16,060
specific values, so this is an example of
discrete random variable. Second is a number
234
00:21:16,060 --> 00:21:19,100
of
road accidents over a particular stretch of
235
00:21:19,100 --> 00:21:25,060
a national highway during a year. So again,
these road accidents can be either 1, 2, 3,
236
00:21:25,060 --> 00:21:31,580
4 in this way up to infinity but, it can never
take any value in between two integers. So,
237
00:21:31,580 --> 00:21:38,360
this also one example of discrete random
variable. Third is the traffic volume at particular
238
00:21:38,360 --> 00:21:43,020
location of a road. So, this is the number
of vehicle that is crossing that particular
239
00:21:43,020 --> 00:21:47,780
section of the road. Again, this can take
only the
240
00:21:47,780 --> 00:21:58,680
integer values for that one. So, this is also
one example of the discrete random variable.
241
00:21:58,680 --> 00:22:04,980
Now, the example of continuous random variable,
on the other end, that amount of rain
242
00:22:04,980 --> 00:22:10,570
received at a particular place over a period
of one year. So, annual rainfall for that
243
00:22:10,570 --> 00:22:14,850
session, for the, say, it is the number of
rainy days, which is a discrete. Now, if I
244
00:22:14,850 --> 00:22:18,100
just say
that total depth of the rain received at that
245
00:22:18,100 --> 00:22:26,990
particular point that can vary from 0 to any
possible number. So, this is one example of
246
00:22:26,990 --> 00:22:32,700
the continuous random variable.
Compressive strength of a concrete cube; now,
247
00:22:32,700 --> 00:22:36,570
there are different ways, there are
different criteria, maintaining different
248
00:22:36,570 --> 00:22:39,940
design criteria, if we prepare the concrete
cube
249
00:22:39,940 --> 00:22:44,919
and then, we test what is this compressive
strength. That can actually take any value
250
00:22:44,919 --> 00:22:51,740
between certain range. So, this one also one
example of the continuous random variable.
251
00:22:51,740 --> 00:22:57,690
Now, coming to the mixed random variable.
So, here say that depth of rainfall at a
252
00:22:57,690 --> 00:23:02,590
particular rain gauge station. So, this depth
of rainfall at a particular rain gauge station,
253
00:23:02,590 --> 00:23:09,460
which is, gives this one, that is as a continuous
random variable here. Now, if you see
254
00:23:09,460 --> 00:23:17,220
that, there if, there are station where most
of the days, if it is the 0 rainfall for the
255
00:23:17,220 --> 00:23:20,440
station,
then how it will take. So, for a 0, suppose
256
00:23:20,440 --> 00:23:27,750
that out of 365 days in a year, suppose that,
there are 220 days are 0 rainfall. So, there
257
00:23:27,750 --> 00:23:30,690
is a probability mass that is concentrated
at a
258
00:23:30,690 --> 00:23:36,590
particular value of depth of rainfall 0 and
for the non zero value, it can have some
259
00:23:36,590 --> 00:23:42,200
distribution, so that distribution should
have shown the first part at particular value
260
00:23:42,200 --> 00:23:45,880
that x
equal to 0, that is the depth equal to 0.
261
00:23:45,880 --> 00:23:49,970
Some mass is concentrated there, some probability
mass is concentrated there and rest of the
262
00:23:49,970 --> 00:23:52,090
probability is distributed over the positive
side
263
00:23:52,090 --> 00:24:02,460
of the axis. So, this is an example of the
mixed random variable.
264
00:24:02,460 --> 00:24:03,460
..
265
00:24:03,460 --> 00:24:10,620
So, saying these three types of random variable,
it is important to note that another
266
00:24:10,620 --> 00:24:18,690
special type of random variable, which is
known as the indicator random variable. So,
267
00:24:18,690 --> 00:24:22,480
it
is a special kind of random variable associated
268
00:24:22,480 --> 00:24:29,880
with the occurrence of an event. So, this
random variable, which is, says that whether
269
00:24:29,880 --> 00:24:36,960
the, a particular event has occurred or not.
Say, an indicated random variable, IA, is
270
00:24:36,960 --> 00:24:41,950
represents all the outcomes in a set A as
1 and
271
00:24:41,950 --> 00:24:49,000
all the outcome outside A as 0. So, it is
like that, whether yes or no. So, if I say
272
00:24:49,000 --> 00:24:51,830
that
again, if I take the road accident, whether
273
00:24:51,830 --> 00:24:56,990
there is a road accident. If there is road
accident, this random variable in fixed in
274
00:24:56,990 --> 00:25:00,679
such a way that is, that functional
correspondence that we are talking about again
275
00:25:00,679 --> 00:25:03,080
here.
That is, it is function is that, it will just
276
00:25:03,080 --> 00:25:05,700
see whether the accident has taken place or
not. If
277
00:25:05,700 --> 00:25:12,230
it is yes, then it will say 1; if it is no,
it will say 0. Say, the each day I am just
278
00:25:12,230 --> 00:25:15,340
taking this,
measurement and just preparing that series,
279
00:25:15,340 --> 00:25:21,620
so that series consist of either 1 or 0. Again
for the rainy days, I will just see every
280
00:25:21,620 --> 00:25:26,370
day whether it is raining or not. I am not
interested in this random variable, if it
281
00:25:26,370 --> 00:25:30,309
is designed in such a way, that it is not
interested,
282
00:25:30,309 --> 00:25:35,760
what is depth of the rainfall. Only, the interested
is that whether it is a rainy day or not.
283
00:25:35,760 --> 00:25:38,860
If
it is not a rainy day, then it will return
284
00:25:38,860 --> 00:25:41,600
0; if it is rainy day, it will return 1. So,
these types
285
00:25:41,600 --> 00:25:46,790
of random variable just indicate the occurrence
of one particular event. Then, the first
286
00:25:46,790 --> 00:25:51,730
case, whether the occurrence of the event
is the accident, another one is the occurrence
287
00:25:51,730 --> 00:25:56,059
of
the rainfall or not. So, if it is yes, that
288
00:25:56,059 --> 00:25:59,120
is, it is within the outcome of the set, that
indicates
289
00:25:59,120 --> 00:26:04,490
.1. If it is outside, then it is indicates
0. So, this kind of random variable is known
290
00:26:04,490 --> 00:26:11,080
as
indicator random variable.
291
00:26:11,080 --> 00:26:12,080
.
292
00:26:12,080 --> 00:26:16,011
Independence of random variable, this is another
important concept to understand. When
293
00:26:16,011 --> 00:26:22,830
we can say that two random variables are independent,
the two random variables R1 and
294
00:26:22,830 --> 00:26:32,580
R2 are independent, if for all X1 X2 belongs
to this real line, there exist that probability
295
00:26:32,580 --> 00:26:37,640
of
R1 equals to X1and the probability of R2 less
296
00:26:37,640 --> 00:26:41,160
than X2 is equals to probability R1 equals
to
297
00:26:41,160 --> 00:26:47,840
X1 multiplied by probability of R2 equals
to X2. Then, if this condition is satisfied,
298
00:26:47,840 --> 00:26:51,800
we say
that these two random variables are independent.
299
00:26:51,800 --> 00:26:58,020
Now, there are other kinds of
independents are also there, which is known
300
00:26:58,020 --> 00:27:00,600
as the mutually independent and there are
k
301
00:27:00,600 --> 00:27:07,270
wise or the pair wise in independence. Suppose,
that there are three, more than two
302
00:27:07,270 --> 00:27:13,780
random variables, which is the R1 R2 R3, if
we will say that these are mutually
303
00:27:13,780 --> 00:27:22,820
independent, if the probability of this particular
one, particular that is R1 equals to X1
304
00:27:22,820 --> 00:27:28,780
and R2 equals to X2 X3 equals to X3 and all
is equals to the multiplication of the
305
00:27:28,780 --> 00:27:41,559
individual probabilities. Then, this can be
told that, these random variables R1 R2 R3
306
00:27:41,559 --> 00:27:48,050
are
mutually independent. Then, there are k wise
307
00:27:48,050 --> 00:27:54,520
independent random variables. Now, out of
this set, suppose there are n random variables
308
00:27:54,520 --> 00:28:02,200
are there and in that, out of that n random
variables, it will say to be k wise independent,
309
00:28:02,200 --> 00:28:06,629
if all the subsets of the k variables are
mutually independent.
310
00:28:06,629 --> 00:28:07,629
..
311
00:28:07,629 --> 00:28:12,000
So, obviously the k is less than n. So, whatever
the total number of random variables are
312
00:28:12,000 --> 00:28:19,230
there, if I now pickup randomly k random variables
and if it satisfies that the mutually
313
00:28:19,230 --> 00:28:24,110
independence among these k random variables,
then we said those n random variable are
314
00:28:24,110 --> 00:28:30,971
k wise independent. Now, if this k becomes
2, then when this number of the variable
315
00:28:30,971 --> 00:28:38,000
reveals 2 out of n, I take the 2, that if
the subsets are in mutually independent. Then,
316
00:28:38,000 --> 00:28:40,669
this
random variable are said to be pair wise independent.
317
00:28:40,669 --> 00:28:46,370
So, what is happening? Out of n
random variables, I am picking up n random;
318
00:28:46,370 --> 00:28:57,400
I am picking up only two random variables.
One small correction here is that, what that
319
00:28:57,400 --> 00:29:01,740
R2 is less than equals to X2 was written;
so it
320
00:29:01,740 --> 00:29:12,789
should be R2 equals to X2 . Now, the most
important thing, why we just learn this
321
00:29:12,789 --> 00:29:15,639
random variable is that probability distribution
of random variables.
322
00:29:15,639 --> 00:29:16,639
..
323
00:29:16,639 --> 00:29:23,600
The distribution function of a random variable
X is a function, which is denoted by this
324
00:29:23,600 --> 00:29:30,299
capital letter F subscript the capital letter
of that random variable, if it is X, then
325
00:29:30,299 --> 00:29:34,610
this a
capital letter of this X with some smaller
326
00:29:34,610 --> 00:29:41,420
case letter, it may or may not be x, which
can
327
00:29:41,420 --> 00:29:47,350
take any letter. So, which is nothing but
this, inside this one, it is nothing but,
328
00:29:47,350 --> 00:29:50,549
as in few
previous slides we have seen that this is
329
00:29:50,549 --> 00:29:57,320
nothing but, the specific value of that random
variable. So, this is denoted as the, this
330
00:29:57,320 --> 00:29:59,559
is the distribution function of the random
variable
331
00:29:59,559 --> 00:30:03,049
X.
Which is nothing but, is the probability of
332
00:30:03,049 --> 00:30:10,640
that random variable X, when it is less than
equals to that specific value of that random
333
00:30:10,640 --> 00:30:15,031
variable. So, this is the way, we define that
distribution function of the random variable
334
00:30:15,031 --> 00:30:22,020
X, which is valid over the region from the
minus infinity to the plus infinity, so, the
335
00:30:22,020 --> 00:30:28,529
entire real axis. So, here again, the general
notation says that the distribution function
336
00:30:28,529 --> 00:30:37,260
of X Y and Z are generally denoted by f
subscript capital x any letter lowercase letter
337
00:30:37,260 --> 00:30:44,409
FX(x), FY(y), FZ(z) respectively. This
variable, that is a lower case letters, that
338
00:30:44,409 --> 00:30:49,490
what I was just telling here, is x inside
y inside z
339
00:30:49,490 --> 00:30:56,130
inside the parentheses can be denoted by any
letter, as these are nothing but, the specific
340
00:30:56,130 --> 00:31:02,169
value of that random variable. This is also
known as the cumulative distribution function,
341
00:31:02,169 --> 00:31:06,880
when in the, most probably in the next class,
we will just discuss about this cumulative
342
00:31:06,880 --> 00:31:11,260
distribution function. This is actually the
cumulative distribution function of the, this
343
00:31:11,260 --> 00:31:14,700
one
of this particular random variable; this will
344
00:31:14,700 --> 00:31:23,179
be discussed along with probability density
function pdf later. So, what we want to tell
345
00:31:23,179 --> 00:31:26,321
is that, for a particular random variable,
this
346
00:31:26,321 --> 00:31:33,090
.random variable over this range, some probabilities
are assigned for the specific range
347
00:31:33,090 --> 00:31:39,390
and that how it is distributed over this real
axis is known as the distribution function
348
00:31:39,390 --> 00:31:42,399
of
that particular random variable.
349
00:31:42,399 --> 00:31:43,399
.
350
00:31:43,399 --> 00:31:50,480
Now, probability distribution of different
types of random variables. Now, we have
351
00:31:50,480 --> 00:31:55,880
discussed about three different types of random
variables; the first is the discrete random
352
00:31:55,880 --> 00:32:01,120
variable. We will see how it is defined that
it is the mathematical function denoted by
353
00:32:01,120 --> 00:32:05,860
generally, for when it is discrete we denote
by this p( x); this is also known as the
354
00:32:05,860 --> 00:32:11,570
probability mass function. We will discuss
this later. So, this is denoted by p( x) that
355
00:32:11,570 --> 00:32:16,740
satisfy the following properties. The first
one is that the probability of any particular
356
00:32:16,740 --> 00:32:23,039
event x, because this is, as it is discrete,
it can any take specific value,. all any means,
357
00:32:23,039 --> 00:32:29,170
that those specific value, which is there
in the feasible sample space, so that particular
358
00:32:29,170 --> 00:32:35,070
specific value is denoted by p( x) which is
nothing but, the probability of the random
359
00:32:35,070 --> 00:32:41,200
variable taking the specific value x, denoted
by either p inside that specific value x or
360
00:32:41,200 --> 00:32:50,799
p
subscript that value, the specific value.
361
00:32:50,799 --> 00:32:56,960
Obviously, from the axioms of this probability,
this p( x) is a nonnegative, for all the real
362
00:32:56,960 --> 00:33:03,850
x, it can either be 0 or greater than 0.
And, the summation of this all this p( x)
363
00:33:03,850 --> 00:33:07,301
over this possible values of x is 1. This,
again
364
00:33:07,301 --> 00:33:14,120
from the axioms of this probability. Though
mathematically, there is no restriction, in
365
00:33:14,120 --> 00:33:18,700
practice, discrete probability distribution
function only defines the integer values.
366
00:33:18,700 --> 00:33:22,680
So,
this is just, when we have also seen in previous
367
00:33:22,680 --> 00:33:26,809
slide, that some example of this discrete
368
00:33:26,809 --> 00:33:31,600
.probability distribution that this is generally
take the integer values. But, it is not
369
00:33:31,600 --> 00:33:39,330
specificmathematically there is no restriction,
and it can take any specific value, am
370
00:33:39,330 --> 00:33:47,710
defining that random variable. Second type
of this random variable is the continuous
371
00:33:47,710 --> 00:33:52,659
random variable. So, this continuous distribution
function for that thing, it is a
372
00:33:52,659 --> 00:33:57,669
mathematical function, which is denoted by
this capital Fsubscript, that random variable
373
00:33:57,669 --> 00:34:02,940
or any lower case letter, particularly this
random variable lower case letter that satisfy
374
00:34:02,940 --> 00:34:04,070
the
following property.
375
00:34:04,070 --> 00:34:05,070
.
376
00:34:05,070 --> 00:34:09,179
The first is that, for any specific value
of this x should be greater than equal to
377
00:34:09,179 --> 00:34:14,429
0. It is
monotonically increasing and continuous function.
378
00:34:14,429 --> 00:34:19,339
So, here it is monotonically
increasing, that means whenever it starts
379
00:34:19,339 --> 00:34:26,019
from and it will go and it will go on increasing
and you know that this can go to maximum.
380
00:34:26,019 --> 00:34:30,889
So, generally for it is defined from 0 to
1. So
381
00:34:30,889 --> 00:34:37,069
this, it can go up, it starts from 0 and go
up to 1 and it increases monotonically and
382
00:34:37,069 --> 00:34:40,450
it is
then continuous function, as this random variable
383
00:34:40,450 --> 00:34:45,419
itself is continuous. So, it is 1 at x is
equals to infinity and 0 at x is equals to
384
00:34:45,419 --> 00:34:48,789
minus infinity, that is this probability at
this x is
385
00:34:48,789 --> 00:34:57,129
1 and probability x minus infinity is 0.
Then, the last thing is this mixed probability
386
00:34:57,129 --> 00:34:59,570
distribution, which is, which can take it,
for
387
00:34:59,570 --> 00:35:06,450
some range it can take that discrete values
and some range can take continuous value.
388
00:35:06,450 --> 00:35:12,430
This also denoted by F(x) and keeping the
all other properties, all other conditions,
389
00:35:12,430 --> 00:35:14,489
that is
the properties are same, which is obviously
390
00:35:14,489 --> 00:35:17,049
greater than equal to 0. Though, with the
391
00:35:17,049 --> 00:35:22,150
.only difference with this continuous is that,
it is monotonically increasing function with
392
00:35:22,150 --> 00:35:27,229
sudden jumps or the steps and again the third
one is again same.
393
00:35:27,229 --> 00:35:34,209
Now this one, this lies the difference between
this continuous probability distribution and
394
00:35:34,209 --> 00:35:40,309
this mixed probability distribution. Now,
if we just see it here, as we are just telling
395
00:35:40,309 --> 00:35:42,670
that
there are at some point, where the probability
396
00:35:42,670 --> 00:35:47,460
is concentrated, now, if I want to just see
that how this cumulated accumulated over time
397
00:35:47,460 --> 00:35:54,849
and so, it starts from here and it will go
and go up to maybe the way it is drawn, it
398
00:35:54,849 --> 00:35:58,059
should be asymptotic to 1. So, this is the
jump
399
00:35:58,059 --> 00:36:01,989
that we are talking about. So, this is the
jump where the probability masses are
400
00:36:01,989 --> 00:36:04,690
concentrated.
Here it is only once, so it can be concentrated
401
00:36:04,690 --> 00:36:09,069
in some other range. Then, there also will
be one jump. So, wherever, the probability
402
00:36:09,069 --> 00:36:14,769
masses concentrated for some specific value,
so they are generally in this distribution
403
00:36:14,769 --> 00:36:23,189
function. We see that type of jump here, which
is the mixed probability distribution.
404
00:36:23,189 --> 00:36:24,189
.
405
00:36:24,189 --> 00:36:33,410
Now this, the concept, the usage of this concept
of this random variable in statistics, it
406
00:36:33,410 --> 00:36:38,519
is
a never ending list, I should say. So, here
407
00:36:38,519 --> 00:36:43,099
are just few examples are given that to
calculate the intervals of the parameters
408
00:36:43,099 --> 00:36:45,799
and to calculate critical region. We will
see this,
409
00:36:45,799 --> 00:36:50,930
what is critical region and this interval
of parameters in the subsequent classes to
410
00:36:50,930 --> 00:36:55,739
determine the reasonable distributional model
for invariant data. Now, this data can be
411
00:36:55,739 --> 00:36:58,069
of
any field of this civil engineering.
412
00:36:58,069 --> 00:37:04,920
.Now, for this analysis, the distributional
model for those kinds of data, this is useful
413
00:37:04,920 --> 00:37:06,690
to
verify the distributional assumptions. We
414
00:37:06,690 --> 00:37:09,209
generally, for any probabilistic model, we
have
415
00:37:09,209 --> 00:37:14,009
assumed some distribution. Now, we have to
verify whether that particular distribution
416
00:37:14,009 --> 00:37:18,209
is
followed or not to study the simulation of
417
00:37:18,209 --> 00:37:21,339
random numbers generated from a specific
probability distribution.
418
00:37:21,339 --> 00:37:22,339
.
419
00:37:22,339 --> 00:37:27,999
Coming to the specific civil engineering,
there are different facets of civil engineering,
420
00:37:27,999 --> 00:37:31,329
starting from water resource engineering,
geotechnical engineering, structural
421
00:37:31,329 --> 00:37:35,480
engineering, transportation engineering, environmental
engineering and there are many
422
00:37:35,480 --> 00:37:40,420
such. So, in water resource, for one example,
the analysis of flood frequency, this
423
00:37:40,420 --> 00:37:47,380
concept is used. In geotechnical engineering,
distribution of in situ stresses in the rock
424
00:37:47,380 --> 00:37:52,690
surrounding and opening or the uniaxial compressive
strength of the rock specimen,
425
00:37:52,690 --> 00:37:56,619
etcetera. In structural engineering, distribution
of the damage stress in a masonry
426
00:37:56,619 --> 00:37:59,980
structure, etcetera, to check the seismic
vulnerability of the structure. Transportation
427
00:37:59,980 --> 00:38:09,440
engineering, for example, the traffic volume
analysis and this kind of thing in different
428
00:38:09,440 --> 00:38:12,839
application of this civil engineering.
429
00:38:12,839 --> 00:38:13,839
..
430
00:38:13,839 --> 00:38:23,269
Now, we will see the concept of the percentile
for a random variable. The u percentile of
431
00:38:23,269 --> 00:38:32,999
a random variable X is the smallest number
Xu, so that u equals to the probability of
432
00:38:32,999 --> 00:38:35,719
X
less than u, which is equals to probability
433
00:38:35,719 --> 00:38:39,609
of Xu. So, to determine what is the value
of this
434
00:38:39,609 --> 00:38:46,200
Xu, the Xu is generally the inverse of the
distribution function F(x), that is, Xu is
435
00:38:46,200 --> 00:38:51,650
equals to
the F(x) inverse u, within the domain 0 to
436
00:38:51,650 --> 00:38:57,079
u to 1.
So now, to know this thing, basically, if
437
00:38:57,079 --> 00:39:00,839
the graphically if I just want to see it here,
now
438
00:39:00,839 --> 00:39:07,099
to show this is the feasible range of that
particular variable, suppose this goes on
439
00:39:07,099 --> 00:39:12,569
here. So
I, now to calculate this percentile, what
440
00:39:12,569 --> 00:39:15,690
some percentile if we just say, some u, so
here
441
00:39:15,690 --> 00:39:23,670
we can say, where is the u. Now, basically,
suppose this is here, so basically, we just
442
00:39:23,670 --> 00:39:26,400
go
and see how much percentage is covered here.
443
00:39:26,400 --> 00:39:31,549
So, this particular value, is graphically
representing this particular value, is your
444
00:39:31,549 --> 00:39:37,630
u percentile of that random variable. So,
generally what happens, we generally see this
445
00:39:37,630 --> 00:39:42,319
particular value, calculate its cumulative
probability and get this one. So, that is
446
00:39:42,319 --> 00:39:45,329
what is your mapping, as this F(x) of any
value
447
00:39:45,329 --> 00:39:49,740
X, that is your mapping. Now, when you are
coming from this side, then what we are
448
00:39:49,740 --> 00:39:59,960
doing, we are just giving this F(x) of u inverse
will give you some value that Xu, which
449
00:39:59,960 --> 00:40:06,359
is your Xu here.
So, this is that inverse function of that
450
00:40:06,359 --> 00:40:12,930
one to get that percentile. So, the u percentile
of a
451
00:40:12,930 --> 00:40:19,039
random variable x is the smallest number Xu
so, that u equals to probability of x less
452
00:40:19,039 --> 00:40:22,390
than
equals to Xu. So, this is obtained that Xu
453
00:40:22,390 --> 00:40:27,109
is equals to inverse of that of the distribution,
of
454
00:40:27,109 --> 00:40:33,319
.that particular number, particular random
variable for that percentile u. And obviously,
455
00:40:33,319 --> 00:40:40,219
the u have the range from 0 to 1 and so it
is expressed in percentages taken from 0 to
456
00:40:40,219 --> 00:40:44,449
1
range, and the range of x being whatever that
457
00:40:44,449 --> 00:40:47,369
range of particular random variable.
.
458
00:40:47,369 --> 00:40:53,979
There are different properties of this distribution
functions. So we will see one by
459
00:40:53,979 --> 00:41:02,160
another, one by one of these properties.
In this discussion, we will follow some notation
460
00:41:02,160 --> 00:41:12,079
this is taken mostly from the Papoulis
book. So, this is that F(x+) of this random
461
00:41:12,079 --> 00:41:22,240
variable X, of course, is equals to limitF(x+ε)
and F(x-) means the limit of this one, when
462
00:41:22,240 --> 00:41:27,449
this are greater than 0 but, it is tending
to 0.
463
00:41:27,449 --> 00:41:34,619
So very small number, so this F(x+), just
right side of that X and F(x-) is the just
464
00:41:34,619 --> 00:41:39,480
left side
of that X. So, property one, the first property
465
00:41:39,480 --> 00:41:43,529
that is, if I take this distribution function
for
466
00:41:43,529 --> 00:41:48,180
this infinity is equals to 1, so that the
right extreme of this real axis and left extreme
467
00:41:48,180 --> 00:41:51,079
of
the real axis, it is starts from 0. So, it
468
00:41:51,079 --> 00:41:55,809
always starts from 0 ends at 1. So, to put
this 1, that
469
00:41:55,809 --> 00:42:03,049
is F plus infinity is equals to F(x) less
than equals to plus infinity. So, if is FX
470
00:42:03,049 --> 00:42:07,049
is less than
equals to plus infinity, that means it is
471
00:42:07,049 --> 00:42:10,470
encompassing the full sample space. So this
is
472
00:42:10,470 --> 00:42:14,869
nothing but, that probability of the full
sample space S and we know that full sample
473
00:42:14,869 --> 00:42:21,930
space from the axioms of the probability,
that this is equals to 1. Similarly, so if
474
00:42:21,930 --> 00:42:24,459
minus
infinity is nothing but, the probability of
475
00:42:24,459 --> 00:42:28,440
x equals to x less than equals to, basically,
less
476
00:42:28,440 --> 00:42:33,480
than equals to minus infinity, which is a,
which is basically, basically null set, so
477
00:42:33,480 --> 00:42:37,759
probability of the null set is equals to 0.
478
00:42:37,759 --> 00:42:38,759
..
479
00:42:38,759 --> 00:42:45,069
Second property,F(x) is a non-decreasing function
of X. That is, this non-decreasing
480
00:42:45,069 --> 00:42:50,410
function or monotonically increasing function,
which is the word that we used in few
481
00:42:50,410 --> 00:42:57,849
slide previous. So, it says that, if that
X1 is less than X2, then always this FX1,
482
00:42:57,849 --> 00:43:04,510
the value of
this distribution function at X1 is less than
483
00:43:04,510 --> 00:43:08,119
equals to FX2. So, it can never decrease.
So, it
484
00:43:08,119 --> 00:43:16,670
will always, it will either be same or it
will increase. So, that is basically what
485
00:43:16,670 --> 00:43:20,789
is known
as this monotonically increasing. Proof of
486
00:43:20,789 --> 00:43:23,940
this one is that, if x(ζ) is less than equal
to x1
487
00:43:23,940 --> 00:43:31,319
and x(ζ) less than equal x2, for some outcome
of this ζ, then X random variable less than
488
00:43:31,319 --> 00:43:41,170
equals to x1 is a subset of the event x less
than equals to x2. So what, so this x1 is
489
00:43:41,170 --> 00:43:44,509
always
there within this x less than equal to x2.
490
00:43:44,509 --> 00:43:50,410
If this is greater than, if this x1 is less
than x2, so
491
00:43:50,410 --> 00:43:55,299
that is why if it is a subset of this one,
then obvious the probability this should be
492
00:43:55,299 --> 00:43:57,970
less
than equals to probability of this x less
493
00:43:57,970 --> 00:44:06,880
than x2.
F(x) increases from 0 to 1. Sorry for this
494
00:44:06,880 --> 00:44:14,079
mistake. F( x) increases from 0 to 1 as x
increases from minus infinity to plus infinity.
495
00:44:14,079 --> 00:44:19,670
This will be 0 to 1 as x increases from
minus infinity to plus infinity.
496
00:44:19,670 --> 00:44:20,670
..
497
00:44:20,670 --> 00:44:34,640
Third property says that, if f x not is equals
to 0, then F(x) equals to 0 for any, which
498
00:44:34,640 --> 00:44:38,499
is
less than this xo. So, for a specific value,
499
00:44:38,499 --> 00:44:42,160
if the F(x) equals to 0, anything which is
lower
500
00:44:42,160 --> 00:44:52,140
than this xo obviously will be 0. This is
basically the same concept of that it is nondecreasing
501
00:44:52,140 --> 00:45:01,479
function. So, if it is some portion, if it
is 0, left side of that in the real access
502
00:45:01,479 --> 00:45:06,849
context that is lower than any value of this
x, obviously that will also be 0. Proof says
503
00:45:06,849 --> 00:45:12,839
since F minus infinity equals to 0, suppose
that x(ζ) is greater than equal to zero for
504
00:45:12,839 --> 00:45:20,839
every ζ. F(0) is equal to probability x less
than equal to 0 as x less than equal to 0
505
00:45:20,839 --> 00:45:26,259
is an
impossible event. So F(x) is 0 for each x
506
00:45:26,259 --> 00:45:31,869
less than xo, this is xo.
.
507
00:45:31,869 --> 00:45:41,680
.Property four, if probability X greater than
x is equal to 1 minus F(x), so probability
508
00:45:41,680 --> 00:45:46,400
X
less than equal to F(x) we designate in this
509
00:45:46,400 --> 00:45:50,819
way. Now we know that total probability is
always 1 so if I just want to know what is
510
00:45:50,819 --> 00:45:54,650
the probability of X greater than x than
obviously this is the rest of this probability
511
00:45:54,650 --> 00:45:59,180
which is 1 minus F(x). Proof says the event
x
512
00:45:59,180 --> 00:46:04,959
less than equals to 1 and x greater than x
are mutually exclusive. So, if these two events
513
00:46:04,959 --> 00:46:11,119
mutually exclusive and collectively exhaustive,
so collectively exhaustive means this x
514
00:46:11,119 --> 00:46:16,319
less than equals to x union x greater than
x is equals to full sample space s. So that,
515
00:46:16,319 --> 00:46:21,260
probability of this less than x and greater
than x, is nothing but, probability of the
516
00:46:21,260 --> 00:46:24,140
total
sample space. That is, the s equals to equal
517
00:46:24,140 --> 00:46:28,049
to 1 and now this is denoted by F(x), which
is
518
00:46:28,049 --> 00:46:34,499
this one equals to 1. So probability of x
greater than x results to 1 minus F(x).
519
00:46:34,499 --> 00:46:35,499
.
520
00:46:35,499 --> 00:46:44,470
The fifth property says, the function F(x)
is continuous from right. So, F(x) plus, that
521
00:46:44,470 --> 00:46:48,989
is
from right is equals to F(x). Proof, since
522
00:46:48,989 --> 00:46:51,979
probability of x less than equals to x plus
ζ
523
00:46:51,979 --> 00:47:00,219
where this ζ is nothing but is tending to
0 is equals to F(x) plus ζ and F(x) plus
524
00:47:00,219 --> 00:47:04,749
ζ is
tending to F(x) plus, when this F, this x
525
00:47:04,749 --> 00:47:09,059
is less than equals to x( ζ) is tending to
F(x) less
526
00:47:09,059 --> 00:47:15,779
than equal to x, as this ζ is tending to
0. So, that is why, this is from this right
527
00:47:15,779 --> 00:47:21,289
hand side,
if we say this is continuous from the right
528
00:47:21,289 --> 00:47:29,319
hand side of any specific value x.
529
00:47:29,319 --> 00:47:30,319
..
530
00:47:30,319 --> 00:47:37,500
Then sixth property says, for a random variable,
if it is bounded by this x1 and x2 , the
531
00:47:37,500 --> 00:47:44,710
probability of the value from starting from
x1 to x2 is equals to the probability of x2
532
00:47:44,710 --> 00:47:46,469
minus x1.
.
533
00:47:46,469 --> 00:47:53,660
So, graphically if I just see it again here,
that is, if I just take two values, if this
534
00:47:53,660 --> 00:47:59,839
is your
say starts form 0 and goes like this. So,
535
00:47:59,839 --> 00:48:03,579
if I want to know that, what if this is your
x1 this
536
00:48:03,579 --> 00:48:11,710
is x2, then, these probabilities that is,
x2, so probability that F(x) less than equals
537
00:48:11,710 --> 00:48:18,330
to x2 is
nothing but, this particular value. Now, probability
538
00:48:18,330 --> 00:48:22,410
of x less than equals to x1 is nothing
539
00:48:22,410 --> 00:48:28,479
.but, this particular value. Now, if I want
to know that, if this x is in between these
540
00:48:28,479 --> 00:48:35,569
two, x2
and x1 then this probability is nothing but,
541
00:48:35,569 --> 00:48:40,059
whatever total probability this minus this
probability. So, this is the probability that
542
00:48:40,059 --> 00:48:42,690
we are talking about, which is nothing but,
the
543
00:48:42,690 --> 00:48:56,390
probability of x1 less than equals to x less
then equals to x2. So, this is how we get
544
00:48:56,390 --> 00:48:58,999
the
probability for a range.
545
00:48:58,999 --> 00:49:06,709
Which are again, this proof says that x less
than equals to x 1 and this x1 less than x
546
00:49:06,709 --> 00:49:10,670
less
than x2 are mutually exclusive. Again, x less
547
00:49:10,670 --> 00:49:20,249
than equals to x2 is equals to x less than
equals to x1 union x1 to x2 So, the probability
548
00:49:20,249 --> 00:49:23,729
x less than equals to x2 is probability x
less
549
00:49:23,729 --> 00:49:32,099
than equals to x 1 plus probability x in between
x1 to x2. Or, if I just take this one here,
550
00:49:32,099 --> 00:49:37,920
then probability of x1, this x random variable
between x1 to x2 is equals to probability
551
00:49:37,920 --> 00:49:41,789
of
x less than x2minus probability of x less
552
00:49:41,789 --> 00:49:44,760
than x1, which is again nothing but, this
F(x1)
553
00:49:44,760 --> 00:49:47,000
minus F(x2).
.
554
00:49:47,000 --> 00:49:57,319
Now, the property seven says, that if probability
of x is equals to a particular value, x is
555
00:49:57,319 --> 00:50:04,369
equals to probability of x minus probability
of x just left to that one. So, at a particular
556
00:50:04,369 --> 00:50:11,259
point, the probability there, now these properties
are in general for the discrete and
557
00:50:11,259 --> 00:50:16,699
continuous. So, for a particular point, the
probability says, that at that particular
558
00:50:16,699 --> 00:50:20,839
point
and just left to that, whatever the probability
559
00:50:20,839 --> 00:50:28,989
is there, so it will be like this; from that
particular side to just to the left of this
560
00:50:28,989 --> 00:50:32,869
one. Proof, putting that x1 equals to x minus
ζ and
561
00:50:32,869 --> 00:50:38,459
x2 equal to x in this property six, then we
can say that probability of x minus ζ less
562
00:50:38,459 --> 00:50:39,459
than
563
00:50:39,459 --> 00:50:45,010
.x less than that x equals to probability
x equals to probability x minus ζxi. Now,
564
00:50:45,010 --> 00:50:48,109
taking
this ζ tending to 0, then we can say that
565
00:50:48,109 --> 00:50:50,579
probability at a particular specific value
is that
566
00:50:50,579 --> 00:50:58,579
point, probability at that point minus immediate
previous value to that one, that particular
567
00:50:58,579 --> 00:51:02,170
value.
So, if it is a discrete random variable, then
568
00:51:02,170 --> 00:51:03,170
just
569
00:51:03,170 --> 00:51:04,170
left
570
00:51:04,170 --> 00:51:05,170
to
571
00:51:05,170 --> 00:51:06,170
this value, this
572
00:51:06,170 --> 00:51:07,170
value
573
00:51:07,170 --> 00:51:11,599
generally comes to 0. So, that for a discrete
value at particular point, the probability
574
00:51:11,599 --> 00:51:16,849
is
equals to that that the functional value at
575
00:51:16,849 --> 00:51:18,150
that particular point.
.
576
00:51:18,150 --> 00:51:26,670
Finally, in this lecture, we have seen that
random variable is not a variable rather a
577
00:51:26,670 --> 00:51:32,920
function which map all the feasible outcome
of an experiment on the real line or a set
578
00:51:32,920 --> 00:51:37,390
of
real numbers. A complex random variable can
579
00:51:37,390 --> 00:51:40,529
only be defined by the joint distribution
of
580
00:51:40,529 --> 00:51:46,849
the real and the imaginary variable, that
is, if that x plus i y will be equal to that,
581
00:51:46,849 --> 00:51:51,380
the joint
distribution of this X and Y, both the random
582
00:51:51,380 --> 00:51:57,759
variables. Random variables can be either
discrete or continuous if the set of events
583
00:51:57,759 --> 00:52:00,539
defined by this variable is either finite
or
584
00:52:00,539 --> 00:52:07,329
infinite numbers respectively. Mixed distributions
are the combination of both discrete
585
00:52:07,329 --> 00:52:13,140
and continuous distribution. Distribution
of random variable with specific application
586
00:52:13,140 --> 00:52:16,769
to
this different civil engineering related problems
587
00:52:16,769 --> 00:52:20,700
will be discussed in the next lecture
along with the concept of this probability
588
00:52:20,700 --> 00:52:23,559
density function and cumulative distribution
function. Thank you.
589
00:52:23,559 --> 00:52:23,559
.