Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.6. Extension to Several Random Variables 139

Example 2.6.2.LetX 1 ,X 2 ,andX 3 be three mutually independent random vari-
ables and let each have the pdf

f(x)=

{

2 x 0 <x< 1

0elsewhere. (2.6.8)

The joint pdf ofX 1 ,X 2 ,X 3 isf(x 1 )f(x 2 )f(x 3 )=8x 1 x 2 x 3 , 0 <xi< 1 ,i=1, 2 ,3,
zero elsewhere. Then, for illustration, the expected value of 5X 1 X 23 +3X 2 X 34 is
∫ 1

0

∫ 1

0

∫ 1

0

(5x 1 x^32 +3x 2 x^43 )8x 1 x 2 x 3 dx 1 dx 2 dx 3 =2.

LetY be the maximum ofX 1 ,X 2 ,andX 3. Then, for instance, we have

P(Y≤^12 )=P(X 1 ≤^12 ,X 2 ≤^12 ,X 3 ≤^12 )

=

∫ 1 / 2

0

∫ 1 / 2

0

∫ 1 / 2

0

8 x 1 x 2 x 3 dx 1 dx 2 dx 3

=(^12 )^6 = 641.

In a similar manner, we find that the cdf ofYis

G(y)=P(Y≤y)=

⎧

⎨

⎩

0 y< 0

y^60 ≤y< 1

11 ≤y.

Accordingly, the pdf ofYis

g(y)=

{

6 y^50 <y< 1

0elsewhere.

Remark 2.6.1.IfX 1 ,X 2 ,andX 3 are mutually independent, they arepairwise
independent(that is,XiandXj,i =j,wherei, j=1, 2 ,3, are independent).
However, the following example, attributed to S. Bernstein, shows that pairwise
independence does not necessarily imply mutual independence. LetX 1 ,X 2 ,andX 3
have the joint pmf

p(x 1 ,x 2 ,x 3 )=

{ 1

4 (x^1 ,x^2 ,x^3 )∈{(1,^0 ,0),(0,^1 ,0),(0,^0 ,1),(1,^1 ,1)}

0elsewhere.

The joint pmf ofXiandXj,i =j,is

pij(xi,xj)=

{ 1

4 (xi,xj)∈{(0,0),(1,0),(0,1),(1,1)}

0elsewhere,

whereas the marginal pmf ofXiis

pi(xi)=

{ 1

2 xi=0,^1

0elsewhere.