November 2001 - Course 1 SOA Solutions

35. E
If X is the random variable representing claim amounts, the probability that X exceeds the
deductible is
∞
[ ] ∫
1
−x −x ∞ −x
Pr X > 1 = xe dx=− xe + e dx (integration by parts)
1 1
−1 −x
∞ −1 −1 −1
1
2
= e − e = e + e = e
= 0.736
It follows that the company expects ( 100)( 0.736)
= 74 claims.
∫
∞
36. C
Let
Then
x = price in excess of 60 that the company charges,
p(x) = price per policy that the company charges,
n(x) = number of policies the company sells per month, and
R(x) = revenue per month the company collects
( ) = 60 + x
( ) = 80 −x
( ) = ( ) ( ) = ( 80 − )( 60 + )
′( ) =− ( 60 + ) + ( 80 − ) = 20 −2
′′( x)
=− 2<
0
p x
n x
R x n x p x x x
R x x x x
R
It follows that R(x) is a maximum when R ( x) 0
20 − 2x
= 0
x = 10
R 10 = 80 − 10 60 + 10 = 4900
and ( ) ( )( )
when R(x) is a maximum.
′ = . We conclude that
Course 1 Solutions 22 November 2001