The Distribution of a Linear Combination
5.5 The Distribution of a Linear Combination
The sample mean and sample total T X o are special cases of a type of random vari-
able that arises very frequently in statistical applications.
DEFINITION
Given a collection of n random variables X 1 , ..., X n and n numerical constants
a 1 , ..., a n , the rv
n
Y5a 1 X 1 1 c1 a n X n 5 g a i X i
i51
is called a linear combination of the X i ’s.
For example, 4X 1 5X 2 8X 3 is a linear combination of X 1 ,X 2 , and X 3 with a 1 4,
a 2 5, and a 3 8.
Taking a 1 a 2 ... a
1 gives Y X 1 ... X n T o , and
1 n
a 1 5a 2 5 c5 a n 5 n yields
X n 5 (X 1 1c1X n )5 T
n
n
Notice that we are not requiring the X i ’s to be independent or identically distrib- uted. All the X i ’s could have different distributions and therefore different mean values and variances. We first consider the expected value and variance of a lin- ear combination.
5.5 The Distribution of a Linear Combination
PROPOSITION
Let X 1 ,X 2 , ..., X n have mean values m 1 , ..., m n , respectively, and variances
s 2 1 2 , c, s n , respectively.
1. Whether or not the X i ’s are independent,
2. If X 1 , ..., X n are independent,
V(a X 1a X 1c1a X )5a 2 1 2 1 2 2 n n 1 V(X 1 )1a 2 V(X 2 ) 1 c1 a n V(X n )
5a 2 s 2 1c1a 2 s 1 2 1 n n
a 1 X 1 1c1a n X n
5 1a 1 s 1 1 c1 a n s n
3. For any X 1 , ..., X n ,
n
n
V(a 1 X 1 1c1a n X n )5 g g a i a j Cov(X i ,X j )
i51 j51
Proofs are sketched out at the end of the section. A paraphrase of (5.8) is that the expected value of a linear combination is the same as the linear combination of the
expected values—for example, E(2X 1 5X 2 ) 2m 1 5m 2 . The result (5.9) in
Statement 2 is a special case of (5.11) in Statement 3; when the X i ’s are independ- ent, Cov(X i , X j )
0 for i ⬆ j and V(X i ) for i j (this simplification actually
occurs when the X i ’s are uncorrelated, a weaker condition than independence). Specializing to the case of a random sample (X i ’s iid) with a i 1n for every i
gives E( ) m and V( ) s X 2 n, as discussed in Section 5.4. A similar comment
applies to the rules for T o .
Example 5.29
A gas station sells three grades of gasoline: regular, extra, and super. These are priced at 3.00, 3.20, and 3.40 per gallon, respectively. Let X 1 , X 2 , and X 3
denote the amounts of these grades purchased (gallons) on a particular day.
Suppose the X i ’s are independent with m 1 1000, m 2 500, m 3 300, s 1 100, s 2 80, and s 3 50. The revenue from sales is Y 3.0X 1 3.2X 2
3.4X 3 , and E(Y) 3.0m 1 3.2m 2 3.4m 3 5620
V(Y) 5 (3.0) 2 s 2 1 (3.2) 2 s 1 (3.4) 2 s 1 2 2 3 5 184,436
s Y 5 1184,436 5 429.46
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The Difference Between Two Random Variables
An important special case of a linear combination results from taking n
2, a 1 1,
and a 2 1: Y a 1 X 1 a 2 X 2 X 1 X 2 We then have the following corollary to the proposition.
CHAPTER 5 Joint Probability Distributions and Random Samples
COROLLARY
E(X 1 X 2 ) E(X 1 ) E(X 2 ) for any two rv’s X 1 and X 2 . V(X 1 X 2 )
V(X 1 ) V(X 2 ) if X 1 and X 2 are independent rv’s.
The expected value of a difference is the difference of the two expected values, but the variance of a difference between two independent variables is the sum, not the
difference, of the two variances. There is just as much variability in X 1 X 2 as in
X 1 X 2 [writing X 1 X 2 X 1 ( 1)X 2 ,( 1)X 2 has the same amount of
variability as X 2 itself].
Example 5.30
A certain automobile manufacturer equips a particular model with either a six-cylinder
engine or a four-cylinder engine. Let X 1 and X 2 be fuel efficiencies for independently and randomly selected six-cylinder and four-cylinder cars, respectively. With m 1 22,
m 2 26, s 1 1.2, and s 2 1.5, E(X 1 X 2 ) m 1 m 2 22 26 4
V(X
1 2X 2 )5s 1 1s 2 5 (1.2)
s X 1 2X 2 5 13.69 5 1.92
If we relabel so that X 1 refers to the four-cylinder car, then E(X 1 X 2 )
4, but the
variance of the difference is still 3.69.
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