Concept of Kurtosis, Business Mathematics & Statistics Video Lecture - Business Mathematics and Statistics - B Com

in this video I want to talk about what
we refer to as ketosis or in other words
the fourth central moment of a
distribution and just thinking about a
particular example I've drawn two random
variables PDFs
and in principle the way in which I've
tried to draw these two variables but it
might be the case that these two random
variables actually have the same variant
and I've also drawn them such that
they're centered around the same value
so perhaps they have a mean of 10 so if
we were to just look at the expected
value of both of these random variables
so the expected value of I where I here
can equal sort of 1 for the first
variable or 2 for the second variable
then the expected value of x I would be
equal to 10 in both cases and similarly
the variance of X I as I sort of intend
to draw it might be the same in both
cases perhaps is equal to 2 let us say
so on the basis of just these first two
centered mode or the first not being a
sense of moment rather second one being
a sense of moment we wouldn't actually
be able to tell these two distributions
apart and that's because we purely sort
of focused on the points that are
relatively near the mean that's what we
kind of deal with the second central
moment how do we then go about looking
at there's no difference between these
two variables in terms of what's going
on in the tails well I'm sure you
probably guessed it I've alluded to it
essentially what we need to do is we
need to look at the full central moment
so the fourth central moment or the it
is referred to as the expectation of X
minus the mean which is 10 to the power
4 and just reminding ourselves of what
this exactly means it means that if we
were to take the process in infinitely
many times and for each sort of
realization of X we then take 10 off and
then we raise that to the power 4 and
then we do that for the second X so X 2
minus n to the power 4 and we just sort
of continue that process forever and
then sort of take the average of all
those processes then that's what it
actually means to have before central
moment and let's think about how it
differ between these two randomvariables
when you sort of have anything raised
the powerful the sort of thing to
realize that as this sort of difference
between X and the mean gets bigger then
when I raise that's the powerful and get
bigger very very quickly if I've got any
sort of large deviations or points away
from the mean then they are going to be
over representative in our sort of
expectations and if we were to calculate
the forth central moment for our second
variable we would find that it was much
much greater than the fourth central
moment for our first variable and that's
because our second variable actually has
points which lie further away from the
mean and it has more of those points
that lie far away from the mean rather
and that's one represented by it sort of
faster tales of its distribution and
because it has more points which lie
sort of reasonable distance away from
the mean when I raise that sort of
distance the power for it means that
this sort of second variable is going to
have a much higher for central moment in
the first and this is kind of what we
refer to as the kurtosis of a
distribution but normally what we do is
we actually refer to the kurtosis of a
distribution we call it sort of gamma
and we say well that's equal to the sort
of forth central moment of our variable
X divided by our sort of standard
deviation all squared so that's now
Sigma to the power four and the reason
we do that is because we're interested
in how spread our sort of variable is
relative to each sort of and standard
deviation because in principle if it was
just this tells you a value of I don't
know a thousand without knowing how
salute and spread our pointer around the
mean so in other other words without
knowing the variant that doesn't really
tell us that much so that's why we kind
of divide through by our standard
deviation or sorry our variance rather
and we square the variance such that
these two things will now have the same
dimensions because we don't want a sort
of measure of a variables kurtosis the
depend on dimensions but then finally
what we do is we take off a value of
three of this and this seems like quite
an arbitrary thing to do and in a sense
it is but the reason we do this is
because this is the ketosis of a normal
distribution and this is this sort of
measure gamma here is sometimes called
kurtosis it's sometimes called excess
kurtosis because it essentially tells us
whether our variable is or has fatter
tails relative to the normal
distribution if gamma is greater than
zero then it has fatter tails if it's
less than zero then it essentially has
thinner tails than a normal distribution
in the next video we're going to talk
about the third central moment of a
distribution I know it kind of seems
like I've done things out of order but
I've done it for a reason the third
central moment of the distribution is a
little bit more complicated to think
about than the fourth central moment