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Hi.
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In this video, we're going to
do standard probability
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calculations for normal
random variables.
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We're given that x is standard
normal with mean 0 and
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variance 1.
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And y is normal with mean
one and variance 4.
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And we're asked for a couple
of probabilities.
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For the normal CDF, we don't
have a closed form expression.
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And so people generally tabulate
values and for the
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standard normal case.
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So if we want little x equal to
3.49, we just look for 3.4
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along the rows and 0.09 along
the columns, and then pick the
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value appropriately.
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So for part A, we're asked
what's the probability that x
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is less than equal to 1.5?
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That's exactly phi of 1.5
and we can look that up.
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1.5 directly and
that's 0.9332.
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Then were asked, what's the
probability that x is less
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than equal to negative 1?
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Notice that negative values
are not on this table.
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And the reason that is is
because the standard normal is
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symmetric around zero.
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And we don't really need that.
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We just recognize that the area
in this region is exactly
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the area in this region.
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And so that's equal to the
probability that x is greater
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than equal to 1.
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This is equal to 1 minus
the probability that x
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is less than 1.
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And we can put the equal sign
in here because x is
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continuous, it doesn't matter.
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And so we're going to get,
this is equal to 1
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minus phi of one.
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And we can look up phi
of 1, which is
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1.00, and that's 0.8413.
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OK.
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For part B, we're asked for
this distribution of y
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minus 1 over 2.
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So any linear function
of a normal random
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variable is also normal.
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And you can see that by using
the derived distribution for
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linear functions of
random variables.
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So in this case, we only need to
figure out what's the mean
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and the variance of this
normal random variable.
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So the mean in this case, I'm
going to write that as y over
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2 minus 1/2.
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The expectation operator
is linear and so
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that's going to be--
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and the expectation in
this case is 1, so
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that's going to be 0.
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Now the variance.
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For the shift, it doesn't
affect the spread.
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And so the variance is exactly
going to be the same without
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the minus 1/2.
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And for the constant, you
can just pull that
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out and square it.
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And the variance of
y we know is 4.
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And so that's 1/4 times
4, that's 1.
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OK.
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So now we know that y
minus 1 over 2 is
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actually standard normal.
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Actually for any normal random
variable, you can follow the
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same procedure.
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You just subtract its mean,
which is 1 in this case.
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And divide by its standard
deviation and you will get a
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standard normal distribution.
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All right, so for part C we want
the probability that y is
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between negative 1 and 1.
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So let's try to massage it so
that we can use the standard
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normal table.
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And we already know that this
is standard normal, so let's
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subtract both sides
by negative 1.
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And that's equal to--
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I'm going to call this standard
normal z, so that's
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easier to write.
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And that's equal to negative 1
less than equal to z, less
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than equal to zero.
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So we're looking for this
region, 0, 1, negative 1.
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So that's just the probability
that it's less than zero minus
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probability that it's less
than negative 1.
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Well for a standard normal, half
the mass is below zero
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and a half the mass is above.
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And so that's just going
to be 0.5 directly.
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And for this, we've already
computed this for a standard
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normal, which was
x in our case.
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And that was 1 minus 0.8413.
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Done.
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So we basically calculated a few
standard probabilities for
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normal distributions.
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And we did that by looking
them up from the standard
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normal table.
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