In this lecture we're going to talk
about mean and variance. So how do we
measure the distribution of continuous
variables? We know that with discrete
variables we can just check how often
your random variable takes on a certain
value and divide it by the total number of
values to give you an approximation, or
an estimation, to that value's probability,
but we can't exactly do that with
continuous variables. We can't really
count them and put them in the buckets like
discrete variables. So we need to measure
certain characteristics of the
distributions that might tell us the
shape of their distribution curve. So two
common measures that you probably have
already heard of are the mean and the
variance, so first let's talk about the
mean. So the mean is like the average. It
means you add up all the different
values and you divide it by the total
number of values, and that kind of tells
you what the middle value would be. So
what I want you to do is to load back up
our random integer CSV.
And so you know how to calculate the sum
of all values, and we want to divide this
by the total number of values. So the
mean of r, so let's just plot r again
to remind ourselves what values that it can
take, so -5 to 5, and so we get an
average value of 0.35. So why is
this value so small, because we generated
values from -5 to 5. With a
uniform distribution we could say that the
distribution is balanced whether the
value is less than 0 or greater than
zero, so a lot of the numbers end up
canceling each other out when you sum
them, and therefore our mean value is
about zero. Notice that we can never
actually get the value .35 from our
distribution right because our
distribution gave us uniformly
distributed numbers between -5
and 5, but only with discrete values.
So you could get negative five, negative
4, negative 3, negative 2, negative 1, or 1
all the way up to 5, but you would
never get the value .35. So it might
seem odd that we also call the mean
value the expected value even though we
would never expect to get that actual
value. We only expect that to be the
average value of the values that we do
draw from this random distribution,
and of course there is an easier way to
calculate the mean in MATLAB and it's
just the function mean. Right, so, we get
the same value. Now let's talk about the
second measure of distribution variance.
So we already talked about the mean
which is kind of like the middle value,
so we could say that it measures the
centeredness of the distribution, so
where the middle is, where the
center is. Variance does a different
thing, it measures the spread. So, one
thing tells us where the random variable
goes and that's the mean, and then the
variance tells us how much it's spread
out from that middle value. So the
definition of the variance is it's the
expected value of the random variable
minus its mean, and then we square that
whole thing, and of course that doesn't
help us much since it doesn't tell us
how to calculate or estimate the
variance, but it's very similar to how we
do the mean. So, let's set the mean of r
to be mean(R) like that. We can then
calculate the variance of R
by taking R minus the mean of R, and
so we want to do the dot product between
these two which is effectively squaring
all the individual values and then
multiplying by the corresponding value
like that. So it's treating R as a
vector, and then finding the dot product
with itself after subtracting the mean.
In other words, it's like subtracting the
mean and then taking the squared distance, or
the squared length, and of course we
divide by the length of R. So we get
10.6475. Of course MATLAB has it's own
variance function, so we didn't really
need to do all this work, and it's just
var. So we get a pretty similar value. One
thing that comes up in statistics which
is a little bit outside of the scope of
our discussion is that if we divide
instead of by n we divide by n minus 1,
that gives us what we call an
unbiased estimate of the variance, and so
this gives us the exact value of
matlab's version of R.