Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.4. Order Statistics 265

4.4.19.LetXandYdenote independent random variables with respective proba-
bility density functionsf(x)=2x, 0 <x<1, zero elsewhere, andg(y)=3y^2 , 0 <
y<1, zero elsewhere. LetU=min(X, Y)andV=max(X, Y). Find the joint pdf
ofUandV.
Hint: Here the two inverse transformations are given byx=u, y =vand
x=v, y=u.

4.4.20.LetthejointpdfofXandYbef(x, y)=^127 x(x+y), 0 <x< 1 , 0 <y<1,
zero elsewhere. LetU=min(X, Y)andV=max(X, Y). Find the joint pdf ofU
andV.

4.4.21.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution of either type.
A measure of spread isGini’s mean difference

G=

∑n

j=2

j∑− 1

i=1

|Xi−Xj|/

(

n

2

)

. (4.4.13)

(a)Ifn= 10, finda 1 ,a 2 ,...,a 10 so thatG=

∑ 10

i=1aiYi,whereY^1 ,Y^2 ,...,Y^10 are

the order statistics of the sample.

(b)Show thatE(G)=2σ/

√

πif the sample arises from the normal distribution

N(μ, σ^2 ).

4.4.22.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of sizen
from the exponential distribution with pdff(x)=e−x, 0 <x<∞, zero elsewhere.

(a)Show thatZ 1 =nY 1 ,Z 2 =(n−1)(Y 2 −Y 1 ),Z 3 =(n−2)(Y 3 −Y 2 ),...,Zn=

Yn−Yn− 1 are independent and that eachZihas the exponential distribution.

(b)Demonstrate that all linear functions ofY 1 ,Y 2 ,...,Yn,suchas

∑n

1 aiYi,can

be expressed as linear functions of independent random variables.

4.4.23.In the Program Evaluation and Review Technique (PERT), we are inter-
ested in the total time to complete a project that is comprised of a large number of
subprojects. For illustration, letX 1 ,X 2 ,X 3 be three independent random times for
three subprojects. If these subprojects are in series (the first one must be completed
before the second starts, etc.), then we are interested in the sumY=X 1 +X 2 +X 3.
If these are in parallel (can be worked on simultaneously), then we are interested in
Z=max(X 1 ,X 2 ,X 3 ). In the case each of these random variables has the uniform
distribution with pdff(x)=1, 0 <x<1, zero elsewhere, find (a) the pdf ofY
and (b) the pdf ofZ.

4.4.24.LetYndenote thenth order statistic of a random sample of sizenfrom
a distribution of the continuous type. Find the smallest value ofnfor which the
inequalityP(ξ 0. 9 <Yn)≥ 0 .75 is true.

4.4.25.LetY 1 <Y 2 <Y 3 <Y 4 <Y 5 denote the order statistics of a random sample
of size 5 from a distribution of the continuous type. Compute: