Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

180 Some Special Distributions

Because theχ^2 -distributions are a subfamily of the Γ-distributions, the additiv-

ity property for Γ-distributions given by Theorem 3.3.1 holds forχ^2 -distributions,

also. Since we often make use of this property, we state it as a corollary for easy

reference.

Corollary 3.3.1.LetX 1 ,...,Xnbe independent random variables. Suppose, for
i=1,...,n, thatXihas aχ^2 (ri)distribution. LetY=

∑n
i=1Xi.ThenY has a
χ^2 (

∑n

i=1ri)distribution.

3.3.2 Theβ-Distribution........................

As we have discussed, in terms of modeling, the Γ-distributions offer a wide vari-
ety of shapes for skewed distributions with support (0,∞). In the exercises and
later chapters, we offer other such families of distributions. How about continuous
distributions whose support is a bounded interval inR? For example suppose the
support ofXis (a, b)where−∞<a<b<∞andaandbare known. Without loss
of generality, for discussion, we can assume thata=0andb= 1, since, if not, we
could consider the random variableY=(X−a)/(b−a). In this section, we discuss
theβ-distributionwhose family offers a wide variety of shapes for distributions
with support on bounded intervals.
One way of defining theβ-family of distributions is to derive it from a pair
of independent Γ random variables. LetX 1 andX 2 be two independent random
variables that have Γ distributions and the joint pdf

h(x 1 ,x 2 )=

1

Γ(α)Γ(β)

xα 1 −^1 xβ 2 −^1 e−x^1 −x^2 , 0 <x 1 <∞, 0 <x 2 <∞,

zero elsewhere, whereα> 0 ,β>0. LetY 1 =X 1 +X 2 andY 2 =X 1 /(X 1 +X 2 ).

We next show thatY 1 andY 2 are independent.

The spaceSis, exclusive of the points on the coordinate axes, the first quadrant

of thex 1 ,x 2 -plane. Now

y 1 =u 1 (x 1 ,x 2 )=x 1 +x 2

y 2 =u 2 (x 1 ,x 2 )=

x 1

x 1 +x 2

may be writtenx 1 =y 1 y 2 ,x 2 =y 1 (1−y 2 ), so

J=

∣

∣

∣

∣

y 2 y 1

1 −y 2 −y 1

∣

∣

∣

∣=−y^1 ≡^0.

The transformation is one-to-one, and it mapsSontoT ={(y 1 ,y 2 ):0<y 1 <
∞, 0 <y 2 < 1 }in they 1 y 2 -plane. The joint pdf ofY 1 andY 2 on its support is

g(y 1 ,y 2 )=(y 1 )

1

Γ(α)Γ(β)

(y 1 y 2 )α−^1 [y 1 (1−y 2 )]β−^1 e−y^1

=

{

y 2 α−^1 (1−y 2 )β−^1

Γ(α)Γ(β) y

α+β− 1

1 e

−y (^10) <y 1 <∞, 0 <y 2 < 1
0elsewhere.