Now the normal distribution has a kurtosis
of three.
It implies a certain tail shape, right?
And so what are some distributions that
have fatter tails than the normal, okay?
So a very common distribution that most
people might have heard of is, is the
Student's t-distribution.
The Student's t-distribution is, is
similar to the standard normal in that
it's a symmetric distribution centered at
zero, But the, student's T distribution
has an extra parameter known as the
degrees of freedom, and, so which I'll
call the degrees of freedom, this Greek
letter Nu.
The degrees of freedom control the tail
thickness of the, the student's T.
And, essentially if this degrees of
freedom parameter is, very large, if in
fact if the, nu is equal to infinity, the
student's t distribution, is, exactly the
normal distribution.
But when the degrees of freedom parameter
is smaller than infinity, then the
student's T distribution has fatter tails
than the normal distribution.
And the smaller the degrees of freedom
parameter, the fatter the tails of the,
student's T distribution.
So this is a, one distribution that is a
candidate for modeling, fat tails, you
know, in, in practice.
That the probability for, for a student to
random variable is like this.
Its, it involves gamma function and the
gamma function is defined here.
Gamma function behaves bit like the
factorial function.
And then but the basic shape of student's
t-distribution is just one + x squared'Y'
divided by the degrees of freedom to the
power of degrees of freedom +one divided
by two.
So, because we are dividing by the degrees
of freedom here, we have to have the
degrees of freedom being positive.
Now, mostly, in, when people work with,
student's T random variables, they treat
the degrees of freedom as integer values.
The degrees of freedom are one, two,
three, or four.
But this distribution is defined for non
integer values of the degrees of freedom.
So the degrees of freedom could be 18.79,
and that's perfectly fine.
So, we'll.
We'll come back to the student's T
distribution later on to, to see at, to
get another definition for what it is.
For right now, it's just a distribution
that has fatter tails than the normal.
And some properties of the student's T
distribution, are as follows.
So if the degrees of freedom parameter is
bigger than one.
Then, the expectation of the random
variable exists and, and is equal to zero.
Okay.
Now the variance of a Student's t random
variable is equal to the degrees of
freedom divided by the degrees of freedom
minus two, and so we need the degrees of
freedom bigger than two in order for the
variance to, to exist.
The skewness is zero and we need the
degrees of freedom actually to be bigger
than three in order to compute the
skewness.
And what's more interesting is that the
kurtosis of a Student's t turns out to be
six divided by the degrees of freedom
minus four minus three.
And, and in order for it to do this
calculation we need the degrees of freedom
to be bigger than four.
>> Right.
>> So, So, if the degrees of freedom were,
five, then we would have, five minus four
is one.
Yeah, so we have six minus three and so
that would be three.
I think this is excess kurtosis.
Yeah, I think that's right.
So in general, whatever the formula should
be is that, we should have a kurtosis
bigger than three for v essentially less
than infinity, and when v is equal to
infinity, the kurtosis should be equal to
three.
Apologize if I think that calculation is,
is not right.
Now, if we want to look at the student's T
random variable, there's a function of R.
There are four functions in R for doing
computations with a student's T.
Rt simulates data from a student's T.
Pt computes the CDF.
Qt computes the quantiles, and DT computes
the density.
Okay?
And all these depend upon this degrees of
freedom parameter, which is the same as
this parameter new.
So from.
To get an idea of, what the, student's
T-distribution looks like, say relative to
the normal.
So here I have different students
t-distributions plotted for degrees of
freedom.
The blue curve is when the degrees of
freedom is 60.
The green curve is when the degrees of
freedom is ten.
The red curve is degrees of freedom five.
And the black curve is, one degree of
freedom.
So notice as the, when the degrees of
freedom get smaller, the, the tails of the
distribution get thicker.
And, and in fact, when, the degrees of
freedom, is around 60, essentially for all
practical purposes, the student's t
distribution is the, standard normal
distribution.
So degrees of freedom doesn't need to go
all the way out to infinity.
So once we get about 60, then the
Student's t and the normal are, are pretty
much the same.
If I drew the normal curve, it would
essentially overlap almost exactly on the
black curve, and there'd be just a tiny
little, sorry, for the blue curve, and the
normal would just deviate from the blue
curve just a tiny bit.