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In this segment and the next
two, we will introduce a few

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useful random variables that
show up in many applications--

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discrete uniform random
variables, binomial random

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variables, and geometric random
variables So let's

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start with a discrete uniform.

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A discrete uniform random
variable is one that has a PMF

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of this form.

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It takes values in a certain
range, and each one of the

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values in that range has
the same probability.

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To be more precise, a discrete
uniform is completely

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determined by two parameters
that are two integers, a and

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b, which are the beginning and
the end of the range of that

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random variable.

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We're thinking of an experiment
where we're going

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to pick an integer at random
among the values that are

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between a and b with the end
points a and b included.

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And all of these values
are equally likely.

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To be more formal, our sample
space is the set of integers

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from a until b.

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And the number of points that we
have in our sample space is

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b minus a plus 1 possible
values.

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What is the random variable
that we're talking about?

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If this is our sample space, the
outcome of the experiment

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is already a number.

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And the numerical value of the
random variable is just the

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number that we happen to
pick in that range.

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So in this context, there isn't
really a distinction

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between the outcome of the
experiment and the numerical

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value of the random variable.

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They are one in the same.

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Now since each one of the values
is equally likely, and

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since we have so many possible
values, this means that the

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probability of any particular
value is going to be 1 over b

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minus a plus 1.

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This is the choice for the
probability that would make

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all the probabilities in
the PMF sum to one.

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What does this random variable
model in the real world?

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It models a case where we have
a range of possible values,

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and we have complete ignorance,
no reason to

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believe that one value is more
likely than the other.

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As an example, suppose that
you look at your digital

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clock, and you look
at the time.

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And the time that it tells you
is 11:52 and 26 seconds.

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And suppose that you just
look at the seconds.

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The seconds reading is something
that takes values in

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the set from 0 to 59.

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So there are 60 possible
values.

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And if you just choose to look
at your clock at a completely

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random time, there's no reason
to expect that one reading

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would be more likely
than the other.

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All readings should be equally
likely, and each one of them

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should have a probability
of 1 over 60.

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One final comment--

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let us look at the special case
where the beginning and

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the endpoint of the range of
possible values is the same,

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which means that our random
variable can only take one

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value, namely that particular
number a.

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In that case, the random
variable that we're dealing

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with is really a constant.

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It doesn't have any
randomness.

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It is a deterministic random
variable that takes a

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particular value of a with
probability equal to 1.

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It is not random in the common
sense of the world, but

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mathematically we can still
consider it a random variable

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that just happens to be the
same no matter what the

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outcome of the experiment is.