November 2001 - Course 1 SOA Solutions

37. A
The joint density of T1 and T
2
is given by
−t
−t
f t , t = e e , t > 0 , t > 0
( )
1 2
1 2 1 2
Therefore,
Pr X ≤ x = Pr 2T + T ≤ x
[ ] [ ]
1 2
⎡
= ∫∫
= −
⎢⎣
1
1
x ( x−t2
) x
( x−t2
)
2 −t1 −t2 −t2 −t1
2
e e dt1dt2 e e dt2
0 0 ∫ ⎢ ⎥
0
0
⎡ ⎤ ⎛ ⎞
= ∫ − = −
⎣ ⎦ ⎝ ⎠
1 1 1 1
x
− x+ t2 x
− x − t
−t
2
2 2 2 −t2
2 2
e ⎢1
e ⎥dt2 ⎜e e e ⎟dt2
0 ∫0
⎡
⎤
= ⎢− e + e e ⎥ =− e + e e + − e
⎣
⎦
1 1 1 1 1
− x − t2
− x − x − x
−t2
2 2 x − x 2 2 2
2
0
2 1 2
1 1
− x
− x
− x −x
2 2 −x
= 1− e + 2e − 2e = 1− 2 e + e , x>
0
It follows that the density of X is given by
1
d ⎡ − x
2 − x
⎤
g ( x)
= ⎢1− 2e + e ⎥
dx ⎣
⎦
1
− x
2
− x
= e − e , x>
0
⎤
⎥⎦
38. E
X , X , and X denote annual loss due to storm, fire, and theft, respectively. In
Let
1 2 3
addition, let Y Max( X , X , X )
Then
= .
1 2 3
[ Y > ] = − [ Y ≤ ] = − [ X ≤ ] [ X ≤ ] [ X ≤ ]
Pr 3 1 Pr 3 1 Pr 3 Pr 3 Pr 3
(
−
e )( −3 e )( −3
e )
(
−
e 3 )(
−
e )( −5
e )
= 1− 1− 1− 1−
= 1− 1− 1− 1−
1 2 3
= 0.414
* Uses that if X has an exponential distribution with mean µ
1
Pr ( X ≤ x) = 1−Pr ( X ≥ x) = 1− ∫ e dt = 1−( − e ) x
= 1−e
µ
∞
x
*
−t µ −t µ ∞ −x
µ
Course 1 Solutions 23 November 2001