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We have already worked through
some examples in which X was a

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random variable with a given
PDF, and we considered the

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problem of finding the PDF of Y
for the case where Y was the

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function x cubed or the function
of the form a/X. What

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both of these examples have
in common is that Y is a

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monotonic function of X.

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In this case, Y is increasing
with X. In this case, Y was

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decreasing with X. It turns out
that there is a general

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formula that gives us the PDF of
Y in terms of the PDF of X

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in the special case where
we're dealing

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with a monotonic function.

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So, let us assume that g is a
strictly increasing function.

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And what that means is that,
if x is a number or smaller

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than some other number x prime,
the value of g of x is

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going to be smaller than
the value of g x prime.

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So, when you increase the
argument of the function, the

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function increases.

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To keep things simple, we will
also assume that the function

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g is smooth, in particular that
it is differentiable.

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Then we have a diagram
such as this one.

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Here is x, and y is given
by a function of x.

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It's a smooth function, and
that function keeps

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increasing.

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Now, because of the assumptions
we have made on g,

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we have an interesting
situation.

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Given a value of x, a
corresponding value of y will

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be determined according
to the function g.

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But we can also go
the other way.

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If I tell you a value of y, then
you can specify for me

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one and only one value of x
that gives rise to this

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particular y.

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So, the function g takes us from
x's to y's, but you can

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also go back the opposite way
from y's to values of x.

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And the mapping that takes us
from y's to x's, this is the

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inverse of the function g.

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And we give a name to that
inverse function,

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and we call it h.

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So, h of y is the value
of x that produces a

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specific value y.

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Let us now move on with the
program of finding the PDF of

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Y. We will follow the usual
two step procedure.

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And the first step is to
find the CDF of Y.

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So we fix some little y,
And we want to find the

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probability that the random
variable y takes a value in

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this range.

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When does this happen?

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For Y to take a value in this
range, it must be the case

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that X takes a value
in this range here.

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Values of X smaller than this
particular number result in

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values of Y that are less
than or equal to

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this particular number.

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So, we can rewrite the event
of interest in terms of the

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random variable X and
write it as follows.

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We need to have x less than
or equal to h of little y.

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But this is just the CDF of
X evaluated at h of y.

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We now carry out to the second
step of our program.

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We take derivatives of both
sides and we find that the PDF

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of Y is equal to the derivative
of the right hand

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side, the derivative of
the CDF is a PDF.

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And then the chain rule tells
us that we also need to take

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the derivative of the term
inside here with respect to

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its argument.

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And this is a general formula
for the PDF of a strictly

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increasing function of a random
variable X. How about

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the case of a decreasing
function?

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So, let us assume that g now
is a strictly decreasing

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function of X.

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So, we might have a plot
for g that looks

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something like this.

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What happens in this case?

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We can start doing a calculation
of this kind.

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But now, how can we rewrite
this event?

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The random variable Y will
take a value less than or

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equal to this number little y.

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When does this happen?

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When the value of g of
x is less than y.

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And that happens for
x's in this range.

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So, this is the set of x's for
which is the value of g of x

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is less than or equal to this
particular number y.

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So the event of interest in that
case is the event that X

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is larger than or equal to h
of y, which is 1 minus the

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probability that X is
less than h of y.

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Because X is a continuous random
variable, we can change

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this inequality to one
that allows the

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possibility of equality.

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And so this is 1 minus the CDF
of X evaluated at h of y.

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Now we take the derivatives of
both sides and we find the PDF

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or Y being equal to, there's
a minus sign here, then the

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derivative of the CDF,
which is the PDF.

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And finally, the derivative
of the function h.

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Now in this case, g is a
decreasing function of x.

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So when x goes down,
y goes up.

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When x goes up, y goes down.

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This means that when y
goes up, x goes down.

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So it means that the inverse
function h is going to be also

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monotonically decreasing.

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Since it is decreasing, it
means that the slope, the

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derivative of the function
h is going to

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be either 0 or negative.

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And so minus a negative value
gives us the absolute value of

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that number.

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So we can rewrite this by
removing this minus sign here,

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and putting an absolute
value in this place.

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Of course, in the case where g
is an increasing function,

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when x goes up, y goes up.

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This means that when y
goes up, x goes up.

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So h in that case would have
been an increasing function,

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so this number here would have
been a non-negative number,

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and so it would be the same
as the absolute value.

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So using these absolute values,
we obtain formulas

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that are exactly the same in
both cases of increasing and

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decreasing functions, and so our
final conclusion is that

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in either case, the PDF of Y is
given in terms of the PDF

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of X times the derivative of
this inverse function.

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Let us now apply the formula
that we have in our hands for

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the monotonic case to a
particular example, where y is

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the square of X, and where
X is uniform on the

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interval 0 to 1.

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So the function g, in our case,
the function g is the

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square function.

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Now, you could argue here that
this function is not

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monotonic, so how can we
apply our results?

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On the other hand, the random
variable X takes values on the

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interval from 0 to 1, and
therefore the form of the

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function g outside that range
does not concern us.

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Over the range of values of
interest, the function g is a

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monotonic function.

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So, what is the correspondence?

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y is going to be equal
to x squared.

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That's the g of x function.

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And when that happens, we have
the relation that x is going

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to be the square root of y.

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This tells us that the inverse
function, h of y, which tells

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us what is the particular x
associated with a given y, the

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inverse function takes the
form square root of y.

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So now we can go ahead
and use the formula.

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The density at some particular
little y where that little y,

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belongs to the range of values
of interest, x things values

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between 0 and 1, so y also takes
values between 0 and 1.

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So over that range, the density
of Y is the density of

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X, which is uniform, therefore
it is equal to 1, times the

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derivative of the square
root function.

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And the derivative of the square
root function is 1 over

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2 times the square root of y.

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As you can see, the amount of
calculations involved here are

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rather simpler compared to what
we would have to do if we

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were to go through our
two step program

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and work with CDFs.

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All that you need to do is
essentially to identify the

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inverse function that given a y
produces x's, and write down

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the corresponding derivative.