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