Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

218 Some Special Distributions

3.6.14.Show that

Y=

1

1+(r 1 /r 2 )W

,

whereWhas anF-distribution with parametersr 1 andr 2 , has a beta distribution.

3.6.15. LetX 1 ,X 2 be iid with common distribution having the pdff(x)=
e−x, 0 <x<∞, zero elsewhere. Show thatZ=X 1 /X 2 has anF-distribution.

3.6.16.LetX 1 ,X 2 ,andX 3 be three independent chi-square variables withr 1 ,r 2 ,
andr 3 degrees of freedom, respectively.

(a)Show thatY 1 =X 1 /X 2 andY 2 =X 1 +X 2 are independent and thatY 2 is

χ^2 (r 1 +r 2 ).

(b)Deduce that

X 1 /r 1

X 2 /r 2

and

X 3 /r 3

(X 1 +X 2 )/(r 1 +r 2 )

are independentF-variables.

3.7 ∗MixtureDistributions..........................

Recall the discussion on the contaminated normal distribution given in Section
3.4.1. This was an example of a mixture of normal distributions. In this section, we
extend this to mixtures of distributions in general. Generally, we use continuous-
type notation for the discussion, but discrete pmfs can be handled the same way.
Suppose that we havekdistributions with respective pdfsf 1 (x),f 2 (x),...,fk(x),
with supports S 1 ,S 2 ,...,Sk,meansμ 1 ,μ 2 ,...,μk, and variances σ 12 ,σ 22 ,...,σ^2 k,
with positive mixing probabilitiesp 1 ,p 2 ,...,pk,wherep 1 +p 2 +···+pk=1. Let
S=∪ki=1Siand consider the function

f(x)=p 1 f 1 (x)+p 2 f 2 (x)+···+pkfk(x)=

∑k

i=1

pifi(x),x∈S. (3.7.1)

Note thatf(x) is nonnegative and it is easy to see that it integrates to one over
(−∞,∞); hence,f(x) is a pdf for some continuous-type random variableX.Inte-
grating term-by-term, it follows that the cdf ofXis:

F(x)=

∑k

i=1

piFi(x),x∈S, (3.7.2)

whereFi(x) is the cdf corresponding to the pdffi(x). The mean ofXis given by

E(X)=

∑k

i=1

pi

∫∞

−∞

xfi(x)dx=

∑k

i=1

piμi=μ, (3.7.3)