WEBVTT
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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.