Fundamentals of Probability and Statistics for Engineers

Verifications of these results are carried out for the case where X 1 ,...,Xn 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,

Proof of Result 4.1:to establish Result 4.1, consider

The first integral in the final expression can be immediately integrated with
respect to x 2 ,x 3 ,...,xn, yielding f 1 (x 1 ), the marginal density function of X 1.
Similarly, the (n 1)-fold integration with respect to x 1 ,x 3 ,...,xn in the second
integral gives fX 2 (x 2 ), and so on. Hence, the foregoing reduces to

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

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

Result 4. 3: let X 1 ,...,Xn be mutually independent random variables. Then
the variance of the sum is the sum of the variances; that is,

94 Fundamentals of Probability and Statistics for Engineers

mYˆm 1 ‡m 2 ‡‡mn: … 4 : 38 †

mYˆEfYgˆEfX 1 ‡X 2 ‡‡Xng

ˆ

Z 1

1

...

Z 1

1

…x 1 ‡‡xn†fX 1 ...Xn…x 1 ;...;xn†dx 1 ...dxn

ˆ

Z 1

1

...

Z 1

1

x 1 fX 1 ...Xn…x 1 ;...;xn†dx 1 ...dxn

‡

Z 1

1

...

Z 1

1

x 2 fX 1 ...Xn…x 1 ;...;xn†dx 1 ...dxn‡...

‡

Z 1

1

...

Z 1

1

xnfX 1 ...Xn…x 1 ;...;xn†dx 1 ...dxn:

X



mYˆ

Z 1

1

x 1 fX 1 …x 1 †dx 1 ‡‡

Z 1

1

xnfXn…xn†dxn

ˆm 1 ‡m 2 ‡‡mn:

Zˆa 1 X 1 ‡a 2 X 2 ‡‡anXn; … 4 : 39 †

mZˆa 1 m 1 ‡a 2 m 2 ‡‡anmn … 4 : 40 †