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Now the normal distribution has a kurtosis
of three.

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It implies a certain tail shape, right?
And so what are some distributions that

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have fatter tails than the normal, okay?
So a very common distribution that most

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people might have heard of is, is the
Student's t-distribution.

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The Student's t-distribution is, is
similar to the standard normal in that

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it's a symmetric distribution centered at
zero, But the, student's T distribution

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has an extra parameter known as the
degrees of freedom, and, so which I'll

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call the degrees of freedom, this Greek
letter Nu.

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The degrees of freedom control the tail
thickness of the, the student's T.

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And, essentially if this degrees of
freedom parameter is, very large, if in

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fact if the, nu is equal to infinity, the
student's t distribution, is, exactly the

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normal distribution.
But when the degrees of freedom parameter

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is smaller than infinity, then the
student's T distribution has fatter tails

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than the normal distribution.
And the smaller the degrees of freedom

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parameter, the fatter the tails of the,
student's T distribution.

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So this is a, one distribution that is a
candidate for modeling, fat tails, you

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know, in, in practice.
That the probability for, for a student to

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random variable is like this.
Its, it involves gamma function and the

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gamma function is defined here.
Gamma function behaves bit like the

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factorial function.
And then but the basic shape of student's

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t-distribution is just one + x squared'Y'
divided by the degrees of freedom to the

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power of degrees of freedom +one divided
by two.

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So, because we are dividing by the degrees
of freedom here, we have to have the

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degrees of freedom being positive.
Now, mostly, in, when people work with,

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student's T random variables, they treat
the degrees of freedom as integer values.

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The degrees of freedom are one, two,
three, or four.

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But this distribution is defined for non
integer values of the degrees of freedom.

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So the degrees of freedom could be 18.79,
and that's perfectly fine.

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So, we'll.
We'll come back to the student's T

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distribution later on to, to see at, to
get another definition for what it is.

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For right now, it's just a distribution
that has fatter tails than the normal.

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And some properties of the student's T
distribution, are as follows.

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So if the degrees of freedom parameter is
bigger than one.

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Then, the expectation of the random
variable exists and, and is equal to zero.

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Okay.
Now the variance of a Student's t random

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variable is equal to the degrees of
freedom divided by the degrees of freedom

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minus two, and so we need the degrees of
freedom bigger than two in order for the

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variance to, to exist.
The skewness is zero and we need the

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degrees of freedom actually to be bigger
than three in order to compute the

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skewness.
And what's more interesting is that the

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kurtosis of a Student's t turns out to be
six divided by the degrees of freedom

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minus four minus three.
And, and in order for it to do this

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calculation we need the degrees of freedom
to be bigger than four.

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>> Right.
>> So, So, if the degrees of freedom were,

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five, then we would have, five minus four
is one.

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Yeah, so we have six minus three and so
that would be three.

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I think this is excess kurtosis.
Yeah, I think that's right.

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So in general, whatever the formula should
be is that, we should have a kurtosis

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bigger than three for v essentially less
than infinity, and when v is equal to

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infinity, the kurtosis should be equal to
three.

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Apologize if I think that calculation is,
is not right.

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Now, if we want to look at the student's T
random variable, there's a function of R.

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There are four functions in R for doing
computations with a student's T.

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Rt simulates data from a student's T.
Pt computes the CDF.

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Qt computes the quantiles, and DT computes
the density.

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Okay?
And all these depend upon this degrees of

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freedom parameter, which is the same as
this parameter new.

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So from.
To get an idea of, what the, student's

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T-distribution looks like, say relative to
the normal.

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So here I have different students
t-distributions plotted for degrees of

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freedom.
The blue curve is when the degrees of

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freedom is 60.
The green curve is when the degrees of

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freedom is ten.
The red curve is degrees of freedom five.

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And the black curve is, one degree of
freedom.

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So notice as the, when the degrees of
freedom get smaller, the, the tails of the

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distribution get thicker.
And, and in fact, when, the degrees of

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freedom, is around 60, essentially for all
practical purposes, the student's t

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distribution is the, standard normal
distribution.

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So degrees of freedom doesn't need to go
all the way out to infinity.

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So once we get about 60, then the
Student's t and the normal are, are pretty

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much the same.
If I drew the normal curve, it would

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essentially overlap almost exactly on the
black curve, and there'd be just a tiny

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little, sorry, for the blue curve, and the
normal would just deviate from the blue

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curve just a tiny bit.