We have already worked through some examples in which X was a
random variable with a given PDF, and we considered the
problem of finding the PDF of Y for the case where Y was the
function x cubed or the function of the form a/X. What
both of these examples have in common is that Y is a
monotonic function of X.
In this case, Y is increasing with X. In this case, Y was
decreasing with X. It turns out that there is a general
formula that gives us the PDF of Y in terms of the PDF of X
in the special case where we're dealing
with a monotonic function.
So, let us assume that g is a strictly increasing function.
And what that means is that, if x is a number or smaller
than some other number x prime, the value of g of x is
going to be smaller than the value of g x prime.
So, when you increase the argument of the function, the
function increases.
To keep things simple, we will also assume that the function
g is smooth, in particular that it is differentiable.
Then we have a diagram such as this one.
Here is x, and y is given by a function of x.
It's a smooth function, and that function keeps
increasing.
Now, because of the assumptions we have made on g,
we have an interesting situation.
Given a value of x, a corresponding value of y will
be determined according to the function g.
But we can also go the other way.
If I tell you a value of y, then you can specify for me
one and only one value of x that gives rise to this
particular y.
So, the function g takes us from x's to y's, but you can
also go back the opposite way from y's to values of x.
And the mapping that takes us from y's to x's, this is the
inverse of the function g.
And we give a name to that inverse function,
and we call it h.
So, h of y is the value of x that produces a
specific value y.
Let us now move on with the program of finding the PDF of
Y. We will follow the usual two step procedure.
And the first step is to find the CDF of Y.
So we fix some little y, And we want to find the
probability that the random variable y takes a value in
this range.
When does this happen?
For Y to take a value in this range, it must be the case
that X takes a value in this range here.
Values of X smaller than this particular number result in
values of Y that are less than or equal to
this particular number.
So, we can rewrite the event of interest in terms of the
random variable X and write it as follows.
We need to have x less than or equal to h of little y.
But this is just the CDF of X evaluated at h of y.
We now carry out to the second step of our program.
We take derivatives of both sides and we find that the PDF
of Y is equal to the derivative of the right hand
side, the derivative of the CDF is a PDF.
And then the chain rule tells us that we also need to take
the derivative of the term inside here with respect to
its argument.
And this is a general formula for the PDF of a strictly
increasing function of a random variable X. How about
the case of a decreasing function?
So, let us assume that g now is a strictly decreasing
function of X.
So, we might have a plot for g that looks
something like this.
What happens in this case?
We can start doing a calculation of this kind.
But now, how can we rewrite this event?
The random variable Y will take a value less than or
equal to this number little y.
When does this happen?
When the value of g of x is less than y.
And that happens for x's in this range.
So, this is the set of x's for which is the value of g of x
is less than or equal to this particular number y.
So the event of interest in that case is the event that X
is larger than or equal to h of y, which is 1 minus the
probability that X is less than h of y.
Because X is a continuous random variable, we can change
this inequality to one that allows the
possibility of equality.
And so this is 1 minus the CDF of X evaluated at h of y.
Now we take the derivatives of both sides and we find the PDF
or Y being equal to, there's a minus sign here, then the
derivative of the CDF, which is the PDF.
And finally, the derivative of the function h.
Now in this case, g is a decreasing function of x.
So when x goes down, y goes up.
When x goes up, y goes down.
This means that when y goes up, x goes down.
So it means that the inverse function h is going to be also
monotonically decreasing.
Since it is decreasing, it means that the slope, the
derivative of the function h is going to
be either 0 or negative.
And so minus a negative value gives us the absolute value of
that number.
So we can rewrite this by removing this minus sign here,
and putting an absolute value in this place.
Of course, in the case where g is an increasing function,
when x goes up, y goes up.
This means that when y goes up, x goes up.
So h in that case would have been an increasing function,
so this number here would have been a non-negative number,
and so it would be the same as the absolute value.
So using these absolute values, we obtain formulas
that are exactly the same in both cases of increasing and
decreasing functions, and so our final conclusion is that
in either case, the PDF of Y is given in terms of the PDF
of X times the derivative of this inverse function.
Let us now apply the formula that we have in our hands for
the monotonic case to a particular example, where y is
the square of X, and where X is uniform on the
interval 0 to 1.
So the function g, in our case, the function g is the
square function.
Now, you could argue here that this function is not
monotonic, so how can we apply our results?
On the other hand, the random variable X takes values on the
interval from 0 to 1, and therefore the form of the
function g outside that range does not concern us.
Over the range of values of interest, the function g is a
monotonic function.
So, what is the correspondence?
y is going to be equal to x squared.
That's the g of x function.
And when that happens, we have the relation that x is going
to be the square root of y.
This tells us that the inverse function, h of y, which tells
us what is the particular x associated with a given y, the
inverse function takes the form square root of y.
So now we can go ahead and use the formula.
The density at some particular little y where that little y,
belongs to the range of values of interest, x things values
between 0 and 1, so y also takes values between 0 and 1.
So over that range, the density of Y is the density of
X, which is uniform, therefore it is equal to 1, times the
derivative of the square root function.
And the derivative of the square root function is 1 over
2 times the square root of y.
As you can see, the amount of calculations involved here are
rather simpler compared to what we would have to do if we
were to go through our two step program
and work with CDFs.
All that you need to do is essentially to identify the
inverse function that given a y produces x's, and write down
the corresponding derivative.