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Hey guys.
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Welcome back.
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Today, we're going to be working
on a problem that asks
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you to find the PMF of a
function of a random variable.
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So let's just jump right in.
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The problem statement gives
you the PMF for a random
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variable called x.
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So we're told that there's this
random variable x that
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takes on values minus 3, minus
2, minus 1, 1, 2, and 3.
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And for each of those values,
the probability mass lying
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over that value is given
by this formula, x
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squared over a.
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Now I didn't write it here to
save room, but we're also told
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that a is a real number that
is greater than 0.
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And we're told that the
probability of x taking on any
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value outside of the set is 0.
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Now we're asked to do two
things in the problem.
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First is to find the value
of the parameter a.
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And that's sort of a natural
question to ask, because if
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you think about it, the PMF
isn't fully specified.
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And in fact, if you plug in the
wrong number for a, you
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actually won't get
a valid PMF.
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So we'll explore that idea
in the first part.
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And then the second part, you're
given a new random
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variable called z.
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And z happens to be
a function of x.
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In fact, it's equal
to x squared.
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And then you're asked
to compute that PMF.
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So this problem is a good
practice problem.
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I think, at this point, you
guys are sort of newly
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acquainted with the
idea of a PMF, or
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probability mass function.
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So this problem will hopefully
help you get more familiar
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with that concept and how
to manipulate PMFs.
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And by the way, just to make
sure we're all on the same
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page, what does a PMF
really tell you?
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So p sub X, where this is
a capital X, because the
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convention in this class
is to use capital
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letters for random variables.
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So p X of k, this is defined
to be the probability that
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your random variable X takes
on a value of k.
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So essentially, this says--
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and this is just some number.
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So in our particular case,
this would be equal to k
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squared over a.
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And how you can interpret
this is this px guy is
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sort of like a machine.
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He takes in some value that
your random variable could
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take on, and then he spits out
the amount of probability mass
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lying over that value.
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OK.
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So now that we've done that
quick recap, let's get back to
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the first part of the problem.
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So we have this formula for
px of x, and we need
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to solve for a.
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So in order to do that, we're
going to use one of our axioms
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of probability to set
up an equation.
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And then we can solve
precisely for a.
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So namely, we know that every
PMF must sum to 1.
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And so essentially, if you sum
this guy over all possible
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values of x, you should get a 1,
and that equation will let
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us solve for a.
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So let's do that.
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Summation over x of px of x.
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So here, essentially
you're only summing
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over these six values.
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So this is equal to px of minus
3, plus px of minus 2,
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plus px of minus 1, et cetera.
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Oops. px of 2 plus px of 3.
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OK.
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And again, like the
interpretation as we said,
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this number here should be
interpreted as the amount of
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probability mass lying
over minus 3.
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And to help you visualize this,
actually, before we go
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further with the computation,
let's actually plot this PMF.
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So the amount of probability
mass lying over minus 3, the
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way we figure that out is we
take minus 3 and we plug it
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into this formula up here.
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So you get 9/a.
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Now you can do this
for minus 2.
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You've got 4/a, looking
at the formula.
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For 1, you get 1/a.
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And of course, this graph, it's
the mirror image over 0,
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because of the symmetry.
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So hopefully this little
visualization helps you
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understand what I'm
talking about.
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And now we can just read
these values off of the
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plot we just made.
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So we know px minus 3
is equal to px of 3.
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So we can go ahead and just
take 2 times 9/a.
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Similarly, we get 2 times 4/a,
and then plus 2 times 1/a.
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So now it's just a question
of algebra.
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So simplifying this, you're
going to get 18 plus 8 plus 2,
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divided by a.
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And this gives you 28/a.
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And as I argued before, you
know that if you sum a PMF
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over all possible values,
you must get 1.
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So this is equal to 1, which
of course implies that a is
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equal to 28.
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So what we've shown here is that
you actually don't have a
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choice for what value
a can take on.
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It must take on 28.
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And in fact, if you plug in
any other value than 28 in
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here, you actually are not going
to have a valid PMF,
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because it's not going
to sum to 1.
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OK.
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So I'm going to write my answer
here, and then erase to
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give myself more room for part