Krzys’ Ostaszewski: http://www.math.ilstu.edu/krzysio/
Author of the “Been There Done That!” manual for Course P/1
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Exercise for May 12, 2007
An insurance policy has a deductible of 10. Losses follow a probability distribution with
density f X ( x ) = xe! x for x > 0 and f X ( x ) = 0 otherwise. Find the expected payment.
A. e!10
B. 2e!10
C. 10e!10
D. 12e!10
E. 100e!10
Solution.
Let X be the random variable describing losses, and let Y be the amount of payment. Then
X ! 10,
# 0,
Y =$
% X " 10, X > 10.
We can calculate the expected value of Y directly
+"
+"
+"
v = !e! x
=
E (Y ) = # ( x ! 10 ) xe dx = # x e dx ! # 10xe dx =
du = 2xdx dv = e! x dx
10
10
10
!####"####
$
!x
2 !x
u = x2
!x
Integration by parts for the first integral
(
2 !x
= !x e
=
u = 8x
)
x$"
x =10
+"
+
# 2xe
!x
10
v = !e! x
!x
du = 8dx dv = e dx
!###
#"####
$
Integration by parts for the first integral
+"
dx !
# 10xe
!x
dx = 100e
!10
+"
!
10
# 8xe
!x
dx =
10
(
= 100e!10 + 8xe! x
)
x$"
x =10
+"
!8
#e
!x
dx
10
!"
# #
$
=
Survival function of
exponential distribution
with hazard rate evaluated
at x =10
= 100e!10 ! 80e!10 ! 8e!10 = 12e!10 .
Answer D. We could also do this problem by applying the Darth Vader Rule. We have
+(
if y = 0, %
" Pr ( X > 10 ) ,
sY ( y ) = Pr (Y > y ) = #
=
Pr
X
>
y
+
10
=
xe! x dx, =
(
)
&
)
Pr
X
!
10
>
y
,
if
y
>
0,
)
$ (
'
10 + y
=
u = !x
v = e! x
= !xe! x
!x
dx = !dx dv = !e
!###"###$
+(
x*(
+
x =10 + y
Integration by parts
)
e! x dx =
10 + y
!
#"#
$
Survival function
of exponential
with hazard rate 1
evaluated at 10 + y
= (10 + y ) e!10 ! y + e!10 ! y = (11 + y ) e!10 ! y
for any nonnegative y. As the payment random variable is non-negative almost surely, the
expected payment is
+"
# (11 + y ) e
0
!10 ! y
dy = 11e
!10
+"
$
#e
!y
dy
0
!"
# #
$
This equals one because
e! y for y>0 is a density
of an exponential random
variable with hazard rate 1
+e
!10
+"
$
# ye
!y
dy
0
!
#"#
$
= 12e!10 .
This equals one
because the integrand
is the density in this
problem.
Answer D.
© Copyright 2007 by Krzysztof Ostaszewski.
All rights reserved. Reproduction in whole or in part without express written
permission from the author is strictly prohibited.
Exercises from the past actuarial examinations are copyrighted by the Society of
Actuaries and/or Casualty Actuarial Society and are used here with permission.