Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

92 Multivariate Distributions

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

1.0

x

y

( x, − 1 −x^2 )

( x, 1 −x^2 )

Region of Integration for Example A.3.1.

Figure 2.1.2:Region of integration for Example 2.1.5. It depicts the integration

with respect toyat a fixed but arbitraryx.

Notice the space of the random vector is the interior of the square with vertices

(0,0),(1,0),(1,1) and (0,1). The marginal pdf ofX 1 is

f 1 (x 1 )=

∫ 1

0

(x 1 +x 2 )dx 2 =x 1 +^12 , 0 <x 1 < 1 ,

zero elsewhere, and the marginal pdf ofX 2 is

f 2 (x 2 )=

∫ 1

0

(x 1 +x 2 )dx 1 =^12 +x 2 , 0 <x 2 < 1 ,

zero elsewhere. A probability likeP(X 1 ≤^12 ) can be computed from eitherf 1 (x 1 )
orf(x 1 ,x 2 ) because

∫ 1 / 2

0

∫ 1

0

f(x 1 ,x 2 )dx 2 dx 1 =

∫ 1 / 2

0

f 1 (x 1 )dx 1 =^38.

Suppose, though, we want to find the probabilityP(X 1 +X 2 ≤1). Notice that
the region of integration is the interior of the triangle with vertices (0,0),(1,0) and