Distribution for Linear Combinations
Proposition
Let X1 , X2 , . . . , Xn have mean values µ1 , µ2 , . . . , µn , respectively, and
variances σ12 , σ22 , . . . , σn2 , respectively.
1.Whether or not the Xi s are independent,
E (a1 X1 + a2 X2 + · · · + an Xn ) = a1 E (X1 ) + a2 E (X2 ) + · · · + an E (Xn )
= a1 µ1 + a2 µ2 + · · · + an µn
2. If X1 , X2 , . . . , Xn are independent,
V (a1 X1 + a2 X2 + · · · + an Xn ) = a12 V (X1 ) + a22 V (X2 ) + · · · + an2 V (Xn )
= a1 σ12 + a2 σ22 + · · · + an σn2
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
1/5
Distribution for Linear Combinations
Proposition (Continued)
Let X1 , X2 , . . . , Xn have mean values µ1 , µ2 , . . . , µn , respectively, and
variances σ12 , σ22 , . . . , σn2 , respectively.
3. More generally, for any X1 , X2 , . . . , Xn
V (a1 X1 + a2 X2 + · · · + an Xn ) =
n X
n
X
ai aj Cov (Xi , Xj )
i=1 j=1
We call a1 X1 + a2 X2 + · · · + an Xn a linear combination of the Xi ’s.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
2/5
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
a. If you take the bus each morning and evening for a week, what is your
total expected waiting time?
b. What is the variance of your total waiting time?
c. What are the expected value and variance of the difference between
morning and evening waiting times on a given day?
d. What are the expected value and variance of the difference between
total morning waiting time and total evening waiting time on a
particular week?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3/5
Distribution for Linear Combinations
Corollary
E (X1 − X2 ) = E (X1 ) − E (X2 ) and, if X1 and X2 are independent,
V (X1 − X2 ) = V (X1 ) + V (X2 ).
Proposition
If X1 , X2 , . . . , Xn are independent, normally distributed rv’s (with possibly
different means and/or variances), then any linear combination of the Xi s
also has a normal distribution. In particular, the difference X1 − X2
between two independent, normally distributed variables is itself normally
distributed.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
4/5
Distribution for Linear Combinations
Example (Problem 62)
Manufacture of a certain component requires three different maching
operations. Machining time for each operation has a normal distribution,
and the three times are independent of one another. The mean values are
15, 30, and 20min, respectively, and the standard deviations are 1, 2, and
1.5min, respectively.
What is the probability that it takes at most 1 hour of machining time to
produce a randomly selected component?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
5/5