Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.7. Continuous Random Variables 57

F(x)

x

1

1

0.5

(0, 0)

Figure 1.7.3:Graph of the cdf of Example 1.7.7.

Example 1.7.8.Reinsurance companies are concerned with large losses because

they might agree, for illustration, to cover losses due to wind damages that are

between $2,000,000 and $10,000,000. Say thatXequals the size of a wind loss in

millions of dollars, and suppose it has the cdf

FX(x)=

{

0 −∞<x< 0

1 −

(

10

10+x

) 3

0 ≤x<∞.

If losses beyond $10,000,000 are reported only as 10, then the cdf of this censored

distribution is

FY(y)=

⎧

⎪⎨

⎪⎩

0 −∞<y< 0

1 −

(

10

10+y

) 3

0 ≤y< 10 ,

110 ≤y<∞,

which has a jump of [10/(10 + 10)]^3 =^18 aty= 10.

EXERCISES

1.7.1.Let a point be selected from the sample spaceC={c:0<c< 10 }.Let

C⊂Cand let the probability set function beP(C)=

∫

C

1
10 dz. Define the random
variableXto beX(c)=c^2. Find the cdf and the pdf ofX.

1.7.2.Let the space of the random variableX beC={x:0<x< 10 }and
letPX(C 1 )=^38 ,whereC 1 ={x:1<x< 5 }. Show thatPX(C 2 )≤^58 ,where
C 2 ={x:5≤x< 10 }.

1.7.3.Let the subsetsC 1 ={^14 <x<^12 }andC 2 ={^12 ≤x< 1 }of the space
C ={x:0<x< 1 }of the random variableXbe such thatPX(C 1 )=^18 and
PX(C 2 )=^12 .FindPX(C 1 ∪C 2 ),PX(C 1 c), andPX(C 1 c∩C 2 c).