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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
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Exercise for February 10, 2007
May 1983 Course 110 Examination, Problem No. 19
Let X1 , X2 , and X 3 be a random sample from a normal distribution with mean µ ! 0
1
and variance ! 2 =
. What are the values of a and b, respectively, in order for
24
L = aX1 + 4 X2 + bX 3 to have a standard normal distribution?
A. a = !2, b = !2
B. a = !2, b = 2
C. a = !1, b = !3
D. a = 2, b = 2
E. Cannot be determined from the given information
Solution.
The random variable L, as a linear combination of independent normal variables, is
normal itself. We have
E ( L ) = aE ( X1 ) + 4E ( X2 ) + bE ( X 3 ) = ( a + 4 + b ) µ.
In order for L to be standard normal, we must have E ( L ) = 0, so that a + b = !4. Also,
a 2 + 16 + b 2
.
24
In order for L to be standard normal, we must have Var ( L ) = 1, so that a 2 + b 2 = 8. This
gives
2
a 2 + ( !a ! 4 ) = 8,
so that
a 2 + a 2 + 8a + 16 = 8,
and
2a 2 + 8a + 8 = 0,
resulting in
2
a 2 + 4a + 4 = ( a + 2 ) = 0,
i.e., a = !2. We also get b = !a ! 4 = 2 ! 4 = !2.
Answer A.
Var ( L ) = a 2Var ( X1 ) + 16Var ( X 2 ) + b 2Var ( X 3 ) =
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