Hi. In this problem, we'll get some
practice working with PDFs and also using PDFs
to calculate CDFs. So the PDF that we're given
in this problem is here. So we have a random variable,
z, which is a continuous random variable. And we're told that the PDF of
this random variable, z, is given by gamma times 1 plus z
squared in the range of z between negative 2 and 1. And outside of this
range, it's 0. All right, so first thing we
need to do and the first part of this problem is we need to
figure out what gamma is because it's not really a
fully specified PDF yet. We need to figure out exactly
what the value gamma is. And how do we do that? Well, we've done analogous
things before for the discrete case. So the tool that we use
is that the PDF must integrate to 1. So in the discrete case, the
analogy was that the PMF had to sum to 1. So what do we know? We know that when you integrate
this PDF from negative infinity to infinity,
fz of z, it has to equal 1. All right, so what
do we do now? Well, we know what
the PDF is-- partially, except for gamma--
so let's plug that in. And the first thing that we'll
do is we'll simplify this because we know that the PDF is
actually only non-zero in the range negative 2 to 1. So instead of integrating from
negative infinity to infinity, we'll just integrate from
negative 2 to 1. And now let's plug in
this gamma times 1 plus z squared dc. And now the rest of the problem
is just applying calculus and integrating this. So let's just go through
that process. So we get z plus 1/3 z cubed
from minus 2 to 1. And now we'll plug
in the limits. And we get gamma, and that's 1
plus 1/3 minus minus 2 plus 1/3 times minus 2 cubed. And then if we add this all up,
you get 4/3 plus 2 plus 8/3, which will give you 6. So what we end up with
in the end is that 1 is equal to 6 gamma. So what does that tell us? That tells us that, in this
case, gamma is 1/6. OK, so we've actually figured
out what this PDF really is. And let's just substitute
that in. So we know what gamma is. So it's 1/6. So from this PDF, we can
calculate anything that we want to. This PDF, basically, fully
specifies everything that we need to know about this
random variable, z. And one of the things that
we can calculate from the PDF is the CDF. So the next part of the
problem asks us to calculate the CDF. So remember the CDF, we use
capital F. And the definition is that you integrate from
negative infinity to this z. And what do you integrate? You integrate the PDF. And all use some dummy variable,
y, here in the integration. So what is it really doing? It's basically just taking the
PDF and taking everything to the left of it. So another way to think about
this-- this is the probability that the random variable
is less than or equal to some little z. It's just accumulating
probability as you go from left to right. So the hardest part about
calculating the CDFs, really, is actually just keeping track
of the ranges, because unless the PDF is really simple, you'll
have cases where the PDF cold be 0 in some ranges and
non-zero in other ranges. And then what you really have
to keep track of is where those ranges are and where you
actually have non-zero probability. So in this case, we actually
break things down into three different ranges because
this PDF actually looks something like this. So it's non-zero between
negative 2 and 1, and it's 0 everywhere else. So then what that means is
that our job is a little simpler because everything to
the left of negative 2, the CDF will be 0 because there's
no probability density to the left. And then everything to the
right of 1, well we've accumulated all the probability
in the PDF because we know that when you integrate
from negative 2 to 1, you capture everything. So anything to the right of
1, the CDF will be 1. So the only hard part is
calculating what the CDF is in this intermediate range, between
negative 2 and 1. So let's do that case first-- so the case of z is between
negative 2 and 1. So what is the CDF
in that case? Well, the definition is to
integrate from negative infinity to z. But we know that everything
to the left of negative 2, there's no probably density. So we don't need to
include that. So we can actually change this
lower limit to negative 2. And the upper limit is
wherever this z is. So that becomes our integral. And the inside is
still the PDF. So let's just plug that in. We know that it's 1/6 1 plus-- we'll make this y squared-- by. And now it's just
calculus again. And in fact, it's more or less
the same integral, so what we get is y plus 1/3 y cubed
from negative 2 to z. Notice the only thing that's
different here is that we're integrating from negative 2 to
z instead of negative 2 to 1. And when we calculate this out,
what we get is z plus 1/3 z cubed minus minus 2 plus 1/3
minus 2 cubed, which gives us 1/6 z plus 1/3 z cubed plus plus
2 plus 8/3 gives us 14/3. So that actually is our CDF
between the range of negative 2 to 1. So for full completeness, let's
actually write out the entire CDF, because there's two
other parts in the CDF. So the first part is that
it's 0 if z is less than negative 2. And it's 1 if z is
greater than 1. And in between, it's this
thing that we've just calculated. So it's 1/6 z plus 1/3 z cubed
plus 14/3 if z is between minus 2 and 1. So that is our final answer. So the main point of this
problem was to drill a little bit more the concepts
of PDFs and CDFs. So for the PDF, the important
thing to remember is that in order to be a valid PDF, the
PDF has to integrate to 1. And you can use that fact to
help you calculate any unknown constants in the PDF. And then to calculate the CDF,
it's just integrating the PDF from negative infinity to
whatever point that you want to cut off at. And the tricky part, as I said
earlier, was really just keeping track of the ranges. In this case, we've broke it
down into three ranges. If we had a slightly more
complicated PDF, then you would have to keep track
of even more ranged. All right, so I hope that
was helpful, and we'll see you next time.
