Krzys’ Ostaszewski: http://www.krzysio.net
Author of the “Been There Done That!” manual for Course P/1
http://smartURL.it/krzysioP (paper) or http://smartURL.it/krzysioPe (electronic)
Instructor for online P/1 seminar: http://smartURL.it/onlineactuary
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If you have questions about these exercises, please send them by e-mail to:
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May 2003 Course 1 Examination, Problem No. 39, also P Sample Exam Questions,
Problem No. 95, and Dr. Ostaszewski’s online exercise posted October 3, 2009
X and Y are independent random variables with common moment generating function
t2
2
M ( t ) = e . Let W = X + Y and Z = Y – X. Determine the joint moment generating
function M ( t1 ,t 2 ) of W and Z.
A. e2t1 + 2t2
2
2
2
t −t
B. e( 1 2 )
2
t +t
C. e( 1 2 )
Solution.
We calculate directly from the definition
(
(
)
E. et1 +t2
2
D. e2t1t2
2
(
)
)
M W ,Z ( t1 , t 2 ) = E et1W +t2 Z = E et1 ( X +Y ) +t2 (Y − X ) = E e(t1 −t2 ) X e(t1 +t2 )Y =
(
)
(
)
1
= E e(t1 −t2 ) X ⋅ E e(t1 +t2 )Y = e 2


1 2
t
= M (t ) = e 2
evaluated at t =t1 −t 2
(t1 −t2 )2
1
e2
(t1 +t2 )2
1
= e2
(t
2
1
) 12 (t
− 2t1t 2 +t 22 +
2
1
+ 2t1t 2 +t 22
)
= et1 +t2 .
1 2
t
= M (t ) = e 2
evaluated at t =t1 +t 2
Answer E.
© Copyright 2009 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.
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