Krzys’ Ostaszewski: http://www.math.ilstu.edu/krzysio/
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Exercise for April 21, 2007
May 1985 Course 110 Examination, Problem No. 5
Let X and Y be random variables with variances 2 and 3, respectively, and covariance
!1. Which of the following random variables has the smallest variance?
A. 2X + Y
B. 2X ! Y
C. 3X ! Y
D. 4X
E. 3Y
Solution.
We have
Var ( 2X + Y ) = 4Var ( X ) + Var (Y ) + 4Cov ( X,Y ) = 8 + 3 ! 4 = 7,
Var ( 2X ! Y ) = 4Var ( X ) + Var (Y ) ! 4Cov ( X,Y ) = 8 + 3 + 4 = 15,
Var ( 3X ! Y ) = 9Var ( X ) + Var (Y ) ! 6Cov ( X,Y ) = 18 + 3 + 6 = 27,
Var ( 4 X ) = 16Var ( X ) = 32,
Var ( 3Y ) = 9Var (Y ) = 27.
Note that when you add two random variables with negative covariance, you reduce the
variance of the sum in relation to the sum of variances, while subtracting them increases
variance. This eliminates answers B and C. Similar consideration eliminates D, as
variance of Y does not overcome reduction from negative covariance. E is eliminated if
we observe that Y, Y, and Y (summands of 3Y) are perfectly correlated, while X, X, and Y
(summands of 2X + Y) are not, and the variance of X is smaller than the variance of Y.
Answer A.
© Copyright 2007 by Krzysztof Ostaszewski.
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