WEBVTT
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A random variable can take
different numerical values
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depending on the outcome
of the experiment.
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Some outcomes are more likely
than others, and similarly
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some of the possible numerical
values of a random variable
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will be more likely
than others.
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We restrict ourselves to
discrete random variables, and
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we will describe these relative
likelihoods in terms
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of the so-called probability
mass function, or PMF for
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short, which gives the
probability of the different
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possible numerical values.
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The PMF is also sometimes called
the probability law or
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the probability distribution of
a discrete random variable.
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Let me illustrate the idea in
terms of a simple example.
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We have a probabilistic
experiment with
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four possible outcomes.
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We also have a probability
law on the sample space.
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And to keep things simple, we
assume that all four outcomes
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in our sample space are
equally likely.
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We then introduce a random
variable that associates a
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number with each possible
outcome as
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shown in this diagram.
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The random variable,
X, can take one of
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three possible values--
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namely 3, 4, or 5.
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Let us focus on one
of those numbers--
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let's say the number 5.
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So let us focus on x
being equal to 5.
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We can think of the event
that X is equal to 5.
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Which event is this?
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This is the event that the
outcome of the experiment led
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to the random variable
taking a value of 5.
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So it is this particular event
which consists of two
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elements, namely a and b.
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More formally, the event that
we're talking about is the set
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of all outcomes for which the
value, the numerical value of
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our random variable, which is
a function of the outcome,
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that numerical value happens
to be equal to 5.
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And in this example it is a set
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consisting of two elements.
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It's a subset of the
sample space.
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So it is an event.
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And it has a probability.
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And that probability we will be
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denoting with this notation.
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And in our case this probability
is equal to 1/2.
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Because we have two outcomes,
each one has probability 1/4.
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The probability of this
event is equal to 1/2.
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More generally, we will be using
this notation to denote
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the probability of the event
that the random variable, X ,
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takes on a particular
value, x.
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This is just a piece of
notation, not a new concept.
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We're dealing with a
probability, and we indicate
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it using this particular
notation.
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More formally, the probability
that we're dealing with is the
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probability, the total
probability, of all outcomes
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for which the numerical value of
our random variable is this
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particular number, x.
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A few things to notice.
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We use a subscript, X, to
indicate which random variable
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we're talking about.
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This will be useful
if we have several
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random variables involved.
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For example, if we have another
random variable on the
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same sample space, Y, then it
would have its own probability
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mass function which would be
denoted with this particular
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notation here.
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The argument of the PMF, which
is x, ranges over the possible
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values of the random variable,
X. So in this sense, here
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we're really dealing
with a function.
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A function that we could
denote just by p with a
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subscript x.
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This is a function as opposed
to the specific
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values of this function.
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And we can produce plots
of this function.
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In this particular example that
we're dealing with, the
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interesting values of
x are 3, 4, and 5.
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And the associated probabilities
are the value of
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5 is obtained with probability
1/2, the value of 4--
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this is the event that the
outcome is c, which has
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probability 1/4.
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And the value of 3 is also
obtained with probability 1/4
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because the value of 3 is
obtained when the outcome is
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d, and that outcome has
probability 1/4.
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So the probability mass function
is a function of an
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argument x.
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And for any x, it specifies
the probability that the
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random variable takes on
this particular value.
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A few more things to notice.
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The probability mass function is
always non-negative, since
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we're talking about
probabilities and
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probabilities are always
non-negative.
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In addition, since the total
probability of all outcomes is
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equal to 1, the probabilities
of the different possible
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values of the random variable
should also add to 1.
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So when you add over all
possible values of x, the sum
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of the associated probabilities
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should be equal to 1.
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In terms of our picture, the
event that x is equal to 3,
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which is this subset of the
sample space, the event that x
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is equal to 4, which is this
subset of the sample space,
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and the event that x is equal to
5, which is this subset of
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the sample space.
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These three events--
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the red, green, and blue--
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they are disjoint, and together
they cover the entire
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sample space.
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So their probabilities
should add to 1.
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And the probabilities of these
events are the probabilities
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of the different values of the
random variable, X. So the
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probabilities of these
different values
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should also add to 1.
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Let us now go through a simple
example to illustrate the
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general method for calculating
the PMF of a
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discrete random variable.
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We will revisit our familiar
example involving two rolls of
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the tetrahedral die.
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And let X be the result of the
first roll, Y be the result of
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the second roll.
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And notice that we're using
uppercase letters.
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And this is because X and
Y are random variables.
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In order to do any probability
calculations, we also need the
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probability law.
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So to keep things simple, let us
assume that every possible
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outcome, there's 16 of them, has
the same probability which
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is therefore 1 over 16 for
each one of the outcomes.
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We will concentrate on a
particular random variable
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defined to be the sum of the
random variables, X and Y. So
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if X and Y both happen to
be 1, then Z will take
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the value of 2.
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If X is 2 and Y is 1 our random
variable will take the
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value of 3.
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And similarly if we have this
outcome, in those outcomes
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here, the random variable
takes the value of 4.
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And we can continue this way
by marking, for each
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particular outcome, the
corresponding value of the
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random variable of interest.
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What we want to do now is to
calculate the PMF of this
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random variable.
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What does it mean to
calculate the PMF?
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We need to find this value for
all choices of z, that is for
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all possible values in the range
of our random variable.
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The way we're going to do it is
to consider each possible
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value of z, one at a time, and
for any particular value find
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out what are the outcomes--
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the elements of the
sample space--
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for which our random variable
takes on the specific value,
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and add the probabilities
of those outcomes.
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So to illustrate this process,
let us calculate the value of
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the PMF for z equal to 2.
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This is by definition the
probability that our random
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variable takes the value of 2.
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And this is an event that
can only happen here.
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It corresponds to only one
element of the sample space,
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which has probability
1 over 16.
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We can continue the same way
for other values of z.
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So for example, the value of PMF
at z equal to 3, this is
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the probability that our random
variable takes the
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value of 3.
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This is an event that can
happen in two ways--
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it corresponds to
two outcomes--
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and so it has probability
2 over 16.
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Continuing similarly, the
probability that our random
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variable takes the value of
4 is equal to 3 over 16.
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And we can continue this way
and calculate the remaining
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entries of our PMF.
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After you are done, you
end up with a table--
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or rather a graph--
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a plot that has this form.
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And these are the values of the
different probabilities
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that we have computed.
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And you can continue with
the other values.
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It's a reasonable guess that
this was going to be 4 over
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16, this is going to be 3 over
16, 2 over 16, and 1 over 16.
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So we have completely determined
the PMF of our
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random variable.
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We have given the form
of the answers.
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And it's always convenient to
also provide a plot with the
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answers that we have.