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We now want to introduce some
examples of random variables,

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and we will start with
the simplest

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conceivable random variable--

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a random variable that takes
the values of 0 or 1, with

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certain given probabilities.

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Such a random variable
is called a

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Bernoulli random variable.

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And the distribution of this
random variable is determined

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by this parameter p, which is
a given number that lies in

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the interval between 0 and 1.

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Using PMF notation, we have
the probability of 0 being

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equal to 1 minus p and the
probability of taking the

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value 1 equal to p.

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If you wish to plot this
particular PMF, the plot is

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rather simple.

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It consists of two bars,
one at 0 and one at 1.

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This one has a height
of p and this has a

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height of 1 minus p.

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Bernoulli random variables show
up whenever you're trying

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to model a situation where
you run a trial.

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And that trial can result in
two alternative outcomes,

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either success or failure, or
heads versus tails, and so on.

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Another situation where
Bernoulli random variables

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show up is when we're making a
connection between events and

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random variables.

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Here's how this connection
is made.

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We have our sample
space, omega.

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And within that sample space,
we have a certain event, A.

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And outside of the event A, of
course, we have the complement

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of A.

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Our random variable is defined
so that it takes a value of 1,

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whenever the outcome of the
experiment lies in A. And it

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takes a value of 0 whenever the
outcome of the experiment

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lies outside the event A,
so that it lies in the

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complement.

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This random variable is called
the indicator random variable

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of the event A. It is
equal to 1 if and

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only if event A occurs.

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And the PMF of that
random variable

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can be found as follows.

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This is PMF notation.

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This is the equivalent
probabilistic notation.

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This is the probability that
the random variable

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takes a value 1.

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Now the random variable takes
the value of 1 if and only if

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event A occurs.

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And so what we have is that
our random variable, the

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indicator random variable, is
a Bernoulli random variable

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with a parameter p equal to the
probability of the event

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of interest.

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Indicator random variables are
very useful because they allow

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us to translate a manipulation
of events to a manipulation of

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random variables.

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And sometimes the algebra of
working with random variable

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is easier than working with
events, as we will see in some

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later examples.