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All of our examples so far have
involved functions, g of

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x, that are monotonic in
X, at least over the

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range of x's of interest.

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Let us now look at an example
that involves a

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non-monotonic function.

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We're going to consider the
square function, which has the

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shape shown in this diagram.

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And we'll assume that X has a
general distribution so that

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it can take both positive
and negative values.

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And so over the range of values
of X, the function that

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we're dealing with is
decreasing and then

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increasing.

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So it is not monotonic.

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How can we find the
distribution of Y?

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As a warm-up, let's look
at the discrete case.

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And as an example of the
calculation, let us find the

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formula for the probability
that the random variable Y

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takes a value of 9.

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This event can happen
in two ways.

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It can happen if X
is equal to 3.

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But it can also happen if x
is equal to negative 3.

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And these are the two and only
two ways that y can take a

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value of 9.

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We can generalize this
calculation.

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The probability that the random
variable y takes on a

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specific value little y, this
probability can be found by

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adding the probabilities of all
of the different x's that

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lead to this particular value.

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Now, for X squared to be equal
to little y, we need to have X

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to be equal either to the
positive square root of y or

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to be equal to the negative
square root of y.

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And this is the general formula
for the PMF of the

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random variable Y. And it
involves two terms, because

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any given value of little y can
happen in two ways, either

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by having X equal to the
negative square root of y or

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by having X be equal to the
positive square root of y.

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We have here a situation where
the function that we're

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dealing with is not
invertible.

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For a given value of y, we
cannot find a single value of

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x that will lead to that y.

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But instead, we typically have
two values of x that lead to

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that particular y, namely the
positive and the negative

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square roots of y.

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So we cannot use the tools that
we used before in the

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monotonic case, where we dealt
with the inverse function.

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What we can do instead is to
proceed from first principles

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and calculate the CDF of the
random variable Y. The CDF of

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Y is the probability that the
random variable is less than

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or equal to a certain number.

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And we're going to focus only on
the case where that certain

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number is non-negative.

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If little y is negative, then we
know that this probability

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is going to be 0, because the
random variable Y cannot take

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negative values.

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Now, this is the probability
that the random variable X

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squared is less than
or equal to y.

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So what we did here is to
express this event in terms of

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the original random variable,
capital X, whose PDF is

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presumably available.

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Now, this event is the same as
requiring the absolute value

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of X to be less than or equal
to the square root of y.

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And this event, again, is the
same as having the random

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variable X be between the
negative and the positive

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square root of y.

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In terms of a picture, the
random variable capital Y

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takes a value less than or
equal to this particular

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little y if and only if the
random variable x falls inside

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this range.

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Now, we want to express this
probability in terms of the

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CDF of X. The probability that
we're looking at, the

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probability of this interval,
is equal to the probability

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that x is less than or equal
to the square root of y.

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This is the probability from
minus infinity up to the

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square root of y.

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But from this, we need to
subtract the probability of

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this interval.

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And that would be the CDF of
the random variable x up to

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the point [negative]

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square root of y.

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00:05:13,040 --> 00:05:17,010
So we now have an expression for
the CDF of Y in terms of

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00:05:17,010 --> 00:05:22,650
the CDF of X. At this point,
now we can take derivatives

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and use the chain rule.

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The PDF of Y is going to be
equal to the derivative of

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this expression.

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The derivative of the first
term, by the chain rule, is

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the PDF of X, evaluated at the
square root of y times the

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derivative of this argument with
respect to y, which is 1

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over 2 square root of y.

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00:05:52,280 --> 00:05:55,770
And then we need the derivative
of the second term.

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We have a minus sign, then
the derivative of the CDF

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which is the PDF.

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Evaluate it at minus
square root of y.

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And then the derivative of this
term with respect to y,

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which is minus 1 over two
square root of y.

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Now, we have a minus sign here
and a minus sign there.

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So the two cancel out.

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We can get rid of this minus 1
term and change this minus

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into a plus.

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00:06:35,020 --> 00:06:38,390
And this is the final
form of the answer.

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So we see that the PDF of Y
evaluated at a particular

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point, which tells us
something about the

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probability that the random
variable takes values around

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this point, has to do with the
probabilities that the random

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00:06:52,490 --> 00:06:58,796
variable X takes values around
here or around there.

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There are two contributions, and
this is because there are

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two different ways that a value
of y may occur, either X

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falling here or X
falling there.