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Hello, welcome to another module in this massive
open online course on probability and random
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variables for wireless communications.
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Today, we are going to start with a new topic
that is the concept of a continuous random
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variables.
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So we are going to start dealing with the
notion of a random variable which is an important
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concept.
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So we will start dealing with a random variable
and a random variable can take basically as
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the name implies, can take values randomly.
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So a random variable, X.
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If X is a random variable, then X can take
values randomly from either the entire set
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of real numbers or a subset of real numbers.
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So X which is a random variable can take values
randomly from entire or subset of real numbers.
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So X is a continuous random variable.
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It can take values randomly from the set of
real numbers.
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And this random variable X is characterized
by a very important function.
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This is known as the probability density function.
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That is, this is characterized by F of X,
the subscript X denotes the random variable.
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The small X denotes the value of this probability
density function at a particular point X.
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So this is the probability density function
also
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denoted by PDF.
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This is the probability density function or
PDF of the random variable X, PDF of the random
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variable
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X.
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And what does this PDF, probability density
function of the random variable X, what does
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it to denote?
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What does the probability density function
of this random variable, X denote?
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Let us consider a small interval of width
dX around X.
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That is let us consider a small interval between
X and X plus dx.
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So let us consider a small interval between
X and let us say this is the PDF.
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Let us consider a small interval, infinitesimally
small interval between X and X plus dX.
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So this is my point X, this is my point X
plus dX.
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We are considering an infinitesimally small
interval of width dX.
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This is my probability density function F
of X of X.
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So we are considering, so consider
an infinitesimal interval X, X plus dX.
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That is we are considering an infinitesimal
interval of width dX around X.
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That is, the interval between X and X plus
dX.
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Then the quantity, F of X times dX, then if
we look at this quantity, F of X times dX,
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this represents, this is the probability
that random variable
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X lies in the interval X and to X plus dX.
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X lies in the interval X to X plus dX.
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So if we look at this infinitesimal interval
between X and X plus dX, the quantity F of
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X times X dX, so the quantity, so the probability
density function is F of X of X. Correct?
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And F of X of X times dX basically at a point
X denotes the probability that the random
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variable, capital X takes value in this infinitesimal
interval of width dX.
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That is, in this infinitesimal interval around
X which is the infinitesimal interval X to
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X plus dX.
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so F of X of X times dX represents the probability
that the random variable takes a value in
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this infinitesimal interval.
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So that is the significance of this probability
density function.
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This is the significance of this probability
density function.
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It denotes the probability but it is not straightforward
that is F of X of X is not the probability,
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F of X of X times dX, that denotes the probability
that it takes a value in the infinitesimal
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interval of width dX around X.
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And naturally, since F of X of X represents
the PDF, represents the probability, it must
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be the case that since probability cannot
be 0, F of X of X has to be greater than or
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equal to 0.
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That is, the PDF, the probability density
function, PDF stands for the probability density
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function, is greater than or equal to 0 for
all values of X.
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At any point X, the probability density function
has to always be greater than or equal to
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0.
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It cannot happen that the probability density
function is negative for a certain value of
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X. Okay?
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So the probability density function represents
the probability density of the random variable
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X which when multiplied by dX gives the probability
that the random variable X lies in the infinitesimal
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interval around small X that is lies in the
infinitesimal interval between X and X plus
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dX.
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Also, this naturally means that, therefore
now the probability that X lies in this interval,
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if we consider any interval, AB, probability
that X lies
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in this interval AB, is basically now if you
look at the probability density function again,
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this is the probability density function,
F of X of X.
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And now let us say, I am considering this
interval between 2 points, A and B. The probability
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that this random variable lies in the interval
A to B is the area under the probability density
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function between the points A and B that is
equal to integral A to B of F of X of X dX.
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Therefore what we are saying is that the probability
that the random variable X lies in this interval
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between A and B is the area under the probability
density function between these 2 points A
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and B.
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Therefore the probability, I can talk about
the probability of X, the random variable
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X which is a continuous random variables that
is it takes values on the real line.
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The probability that X is an element of A
times B is equal to integral of A to B, F
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of X of X dX which is the same thing as saying
that the probability X element of AB is basically
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this quantity if you can look at this, this
quantity A to B of F of X dX is basically
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nothing but the area under the probability
density function.
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This quantity is basically the area under
the PDF that is the probability density function
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between the points A, B.
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So if I look at the interval A, B, the area
under the probability density function between
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these points, A and B that is integral, A
to B of F of X dX represents the probability
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that the random variable X is an element or
belongs to this interval between A and B or
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that the random variable X takes values from
this interval A to B.
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Naturally therefore, it follows that I am
saying naturally because it follows from the
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above result, if we consider integral minus
infinity to infinity, F of X of X dX, that
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must be equal to 1.
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If I consider the integral between the limits
minus infinity to infinity of F of X dX where
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F of X is the probability density function,
this integral over the entire set of real
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numbers must be infinity because the entire
set of real numbers is the sample space.
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It is the sample space for this random variable,
therefore the probability that it takes any
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value from the sample space must be 1.
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The total probability that it takes any value
from the Sample space must be equal to 1.
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So what we are saying is, for this random
variable, X, the sample space equals minus
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infinity to infinity, therefore probability
so what we are asking is this is basically
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integral over minus infinity to infinity F
of X of X dX is basically the probability
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that the random variable X belongs to minus
infinity to infinity which is the same as
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the probability that the random variable,
X belongs to the sample space which is equal
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to 1.
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Because this is basically, this quantity as
we had seen in the axioms of probability before,
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this quantity is basically the probability
of the sample space.
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And from the exempts of probability, that
is from the 2nd axiom of probability, we have
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the probability of the sample space equal
to 1.
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Therefore the total area under the probability
density so we have 2 interesting properties,
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one is that the probability density function
F of X of X is always greater than or equal
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to 0.
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And 2, the area, the integral between minus
infinity to infinity F of X of dX and what
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is this?
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This is the total area under the probability
density function on the entire real line between
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the limits, minus infinity to infinity.
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This total area under the probability density
function must be equal to the total probability
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that is unity.
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So the total area under the probability density
function, what is this quantity?
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This quantity is the total area
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under probability density function.
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The total area under the probability density
function must be equal to 1.
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We are saying that the total area under the
probability density function must be equal
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to 1.
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The probability density function is greater
than or equal to 0, the total area under the
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probability density function is equal to 1.
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And, F of X times dX represents the probability
that the random page table variable X takes
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a value in the infinitesimal interval of width
dX around X.
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That is it lies in the interval between X
to X plus dX.
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And integral A to B of F of X dX is basically
the probability that the random variable takes
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values in the interval A to B. All right?
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So for example, let us take a look at this
simple example now.
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For example, let us take a look at,
consider the PDF, F of X of X equals K e to
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the power of minus AX for X greater than or
equal to 0, that is we are given a PDF which
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is some constant, K times e raised to minus
X for X greater than or equal to 0 where A
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is a known constant.
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A is also termed as the parameter.
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This quantity A is also termed as the parameter.
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A is the parameter of this PDF that is the
probability density function.
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And what we are asked to do is we are asked
to find the value of the unknown constant
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K.
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So we are given a PDF which is K e to the
power of minus AX where A is a known parameter.
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This PDF is valid for X greater than or equal
to 0 and what we are asked as we are asked
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what is the value of this constant, okay?
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And naturally, we can find it as follows.
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We know that the total probability is 1.
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Therefore, it must be the case that if I look
at the total probability, this is defined
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from 0 to infinity.
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So therefore, if I look at the integral zero
to infinity of F of X of X, for a general
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PDF it is minus infinity to infinity.
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Therefore, this must be equal to 1.
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However, this is defined only from 0 to infinity.
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It means that from minus infinity to infinity,
the PDF is 0.
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So, 0 to infinity, K times e power minus AX
dX is equal to 1.
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Now this integral, K times e power minus AX,
you can evaluate this integral in a straightforward
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fashion.
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This is minus K over A e to the power of minus
AX evaluated between the limits, 0 to infinity
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which must be equal to 1.
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It means, now if I substitute these limits
at X equal to infinity, this is e power minus
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AX which is e power minus infinity.
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This is 0.
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Therefore, this is 0 minus of at X equal to
0, e power minus AX is e power minus0 which
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is one.
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So this is 0 minus of minus K by A which is
equal to K by A and this is equal to 1 implies
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our parameter K is equal to A. So we derived
that K by A is equal to 1 which means our
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parameter, remember over PDF is K e power
minus AX.
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Right?
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Using the probability that the PDF must be
integral to 1 that is the area under the PDF
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over the entire real line is 1, we have derived
the condition that K is equal to A.
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Therefore the PDF is given as
F of X of X which is equal to K is equal to
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A e to the power of minus AX for X greater
than or equal to 0.
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So let me write this again clearly over here,
F of X of X equals A e to the power of minus
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AX for X greater than or equal to 0 and this
PDF which is decreasing exponentially, this
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PDF looks like this.
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This PDF for A is constant.
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It is decreasing exponentially.
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This is A e to the power of minus X, this
is X.
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This is.
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F of X of X.
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It is decreasing.
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A e to the power of minus AX is basically
decreasing exponentially.
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It starts with A at X is equal to 0 and it
is decreasing exponentially.
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At X equal to 0, this is basically A e to
the power of minus 0 which is A and it is
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decreasing exponentially, this PDF is also
known as the exponential PDF.
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This is known as an exponential PDF.
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This is one of the standard probability density
functions.
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This is known as an exponential probability
density function.
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So F of X of X equals A e to the power of
minus AX for X greater than or equal to 0.
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This is
the exponential probability density function.
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This is the exponential probability density
function and we have also seen that this A
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is a parameter which characterizes this exponential
probability density function.
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So we are saying that this probability density
function F of X of X equals A e to the power
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of minus AX is the exponential probability
density function.
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It is defined for X greater than or equal
to 0.
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And A is the parameter which characterizes
this exponential probability density function.
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So in this module, we have seen a definition
of the probability density function of a random
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variable, the properties of the probability
density what it represents, the probability
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density function, the properties of this probability
density function and also we have done a simple
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problem involving this exponential probability
density function.
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So let us stop this module here.
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We will look at other aspects in the subsequent
modules.
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Thank you very much