238 Frequently Asked Questions In Quantitative Finance
drawn from a bounded distribution. (The figure shows
a ‘humped’ Weibull, but depending on parameter values
the distribution can be monotonic.)
Mean
a+b
(
c+ 1
c
)
.
Variance
b^2
(
(
c+ 2
c
)
−
(
c+ 1
c
) 2 )
.
Where(·) is the Gamma function.
Student’s t Unbounded above and below. It has three
parameters:a, location;b>0, scale;c>0, degrees of
freedom. Its probability density function is given by
(c+ 1
2
)
b
√
πc
(c
2
)
(
1 +
(x−a
b
) 2
c
)−c+ 21
,
where(·) is the Gamma function. This distribution
represents small-sample drawings from a normal distri-
bution. It is also used for representing equity returns.
Mean
a.
Variance
(
c
c− 2
)
b^2.
Note that thenth moment only exists ifc>n.
Pareto Bounded below, unbounded above. It has two
parameters:a>0, scale;b>0 shape. Its probability
density function is given by
bab
xb+^1
x≥a.
