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Let us now give an example of a
continuous random variable--

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the uniform random variable.

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It is patterned after the
discrete random variable.

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Similar to the discrete case,
we will have a range of

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possible values.

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In the discrete case, these
values would be the integers

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between a and b.

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In the continuous case, any real
number between a and b

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will be possible.

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In the discrete case, these
values were equally likely.

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In the continuous case, at all
points, we have the same

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height for the probability
density function.

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And as a consequence, if we take
two intervals that have

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the same length, then these two
intervals will be assigned

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the same probability.

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Intuitively, uniform random
variables model

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the following situation.

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We know that the numerical value
of the random variable

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will be between a and b.

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But we know nothing more.

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We have no reason to believe
that certain locations are

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more likely than others.

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And in this sense, the uniform
random variable models a

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situation of complete
ignorance.

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By the way, since probabilities
must add to 1,

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the area of this rectangle
must be equal to 1.

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And therefore, the height of
this rectangle has to be 1

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over b minus a, so that
we have a height of 1

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over b minus a.

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We have a length of b minus a.

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So the product of the
two, which is the

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area, is equal to 1.

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Finally, here's a more
general PDF, which

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is piecewise constant.

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One thing to notice is that
this, in particular, tells us

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that PDFs do not have to be
continuous functions.

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They can have discontinuities.

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Of course, for this to be a
legitimate PDF, the total area

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under the curve, which is the
sum of the areas of the

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rectangles that we have here,
must be equal to 1.

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With a piecewise constant PDF,
we can calculate probabilities

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of events fairly easy.

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For example, if you wish to find
the probability of this

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particular interval, which is
going to be the area under the

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curve, that area really consists
of two pieces.

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We find the areas of these two
rectangles, add them up, and

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this gives us the total
probability of

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this particular interval.

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So at this point, our
agenda, moving

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forward, will be twofold.

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First, we will introduce
some interesting

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continuous random variables.

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We just started with the
presentation of the uniform

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random variable.

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And then, we will also go over
all of the concepts and

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results that we have developed
for discrete random variables

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and develop them again for their
continuous counterparts.