Krzys’ Ostaszewski: http://www.math.ilstu.edu/krzysio/
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 Course P/1 online seminar: http://smartURL.it/onlineactuary
Questions? E-mail: [email protected]
Exercise for December 16, 2006
May 1985 Course 110 Examination, Problem No. 11
Let X and Y be independent random variables each with density fT ( t ) =
!" < t < " , and 0 otherwise. If Var ( XY ) =
A. 1
B. 2
C.
4 3
3
1
for
2!
64
, then ! = ?
9
D. 2 2
E.
8 3
3
Solution.
Note that X and Y both have the uniform distribution on the interval ( !" ," ) . Using the
assumption of independence of X and Y we conclude that
64
2
2
2
= Var ( XY ) = E ( XY ) ! ( E ( XY )) = E X 2Y 2 ! ( E ( X ) " E (Y )) =
9
(
)
(
( ) ( ) ! ( E ( X )) " ( E (Y ))
= E X "E Y
2
2
2
)
$ (# ! ( !# ))2
'
#4
2
2
2
=&
+ 0 ) ! 0 "0 = .
12
9
%
(
2
2
This gives ! 4 = 64, and ! = 2 2.
Answer D.
© Copyright 2006 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
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