Frequently Asked Questions In Quantitative Finance

Chapter 3: The Most Popular Probability Distributions 245

Cauchy This distribution is unbounded below and above.

It has two parameters:a, location;b>0 scale. Its prob-

ability density function is given by

1

πb

(

1 +

(x−a

b

) 2 ).

This distribution is rarely used in finance. It does not

have any finite moments, but its mode and median are

botha.

Beta This distribution is bounded below and above.

It has four parameters:a, location of lower limit;b>

alocation of upper limit;c>0andd>0 shape. Its

probability density function is given by

(c+d)

(c)(d)(b−a)c+d−^1

(x−a)c−^1 (b−x)d−^1 , a≤x≤b,

Cauchy

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -3 -2 -1 0 1 2 3 4

a = 0.5

b = 1