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