Page 111 - Fundamentals of Probability and Statistics for Engineers

94 Fundamentals of Probability and Statistics for Engineers

Verifications of these results are carried out for the case where X 1 ,..., X n are
continuous. The same procedures can be used when they are discrete.
Result 4.1: the mean of the sum is the sum of the means; that is,

m Y ˆ m 1 ‡ m 2 ‡ ‡ m n : 4:38†

Proof of Result 4.1: to establish Result 4.1, consider

m Y ˆ EfYgˆ EfX 1 ‡ X 2 ‡ ‡ X n g
Z 1 Z 1
ˆ ... x 1 ‡ ‡ x n †f x 1 ; ... ; x n †dx 1 ... dx n
X 1 ...X n
1 1
1 1
Z Z
ˆ ... x 1 f x 1 ; ... ; x n †dx 1 .. . dx n
X 1 ...X n
1 1
1 1
Z Z
‡ ... x 2 f x 1 ; ... ; x n †dx 1 ... dx n ‡ ...
X 1 ...X n
1 1
Z 1 Z 1
‡ ... x n f x 1 ; ... ; x n †dx 1 ... dx n :
X 1 ...X n
1 1
The first integral in the final expression can be immediately integrated with
respect to x 2 , x 3 ,..., x n , yielding f (x 1 ), the marginal density function of X 1 .
X
1
Similarly, the (n 1)-fold integration with respect to x 1 , x 3 ,..., x n in the second
integral gives f (x 2 ), and so on. Hence, the foregoing reduces to
X 2
Z 1 Z 1
m Y ˆ x 1 f x 1 †dx 1 ‡ ‡ x n f x n †dx n
X 1 X n
1 1
ˆ m 1 ‡ m 2 ‡ ‡ m n :
Combining Result 4.1 with some basic properties of the expectation we
obtain some useful generalizations. For example, in view of the second of
Equations (4.3), we obtain Result 4.2.
Result 4.2: if
Z ˆ a 1 X 1 ‡ a 2 X 2 ‡ ‡ a n X n ; 4:39†

where a 1 , a 2 ,..., a n are constants, then

4:40†
m Z ˆ a 1 m 1 ‡ a 2 m 2 ‡ ‡ a n m n

Result 4.3: let X 1 ,..., X n be mutually independent random variables. Then

the variance of the sum is the sum of the variances; that is,

TLFeBOOK