Frequently Asked Questions In Quantitative Finance

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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.
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