Topic 4-2 Linear combinations and the correlation coefficient
Solved problems
Problem 4-2-1
Bob and John are travelling from city A to city D. Bob decides to take the upper route (through B) whereas John
takes the lower route (through C) as shown in the following figure:
B
T2
T1
D
A
T3
T4
C
The travel times (in hours) between the cities indicated are normally distributed as follows:
T1 ~ N (6, 2)
T2 ~ N (4, 1)
T3 ~ N (5, 3)
T4 ~ N (4, 1)
Although the travel times here generally can be assumed statistically independent, T3 and T4 are dependent with
correlation coefficient 0.8.
(a) What is the probability that John will not arrive in city D within 10 hours? (ans. 0.397)
(b) What is the probability that Bob will arrive in city D earlier than John by at least one hour? (ans. 0.327)
(c) Which route (upper or lower) should be taken if one wishes to minimize the expected travel time from A to
D? Justify. (ans. lower)
Solution:
(a) Let TJ be John’s travel time in (hours); TJ = T3 + T4 with
 TJ = 5 + 4 = 9 (hours), and
 T = [32 + 12 + (2)(0.8)(3)(1)]1/2 = 3.847076812 (hours)
J
Hence
P(TJ > 10 hours) = 1 – P(
T J   TJ
T

J
10  9
)
3.847076812
= 1 – ( 0.259937622) = 1 – 0.602543999
 0.397
(b) Let TB be Bob’s travel time in (hours); TB = T1 + T2 with
TB = 6 + 4 = 10 (hours), and
 T = [22 + 12]1/2 = 5 (hours)
B
Hence
P(TJ – TB > 1) = P(TB – TJ + 1 < 0),
now let R  TB – TJ + 1; R is normal with
R = TB – TJ + 1 = 10 – 9 + 1 = 2,
R = [  TB 2 +  TJ 2]1/2 = (5 + 14.8)1/2 = 19.8 ,
hence
P(R < 0) = P(
R  R
02
)
R
19.8
= (-0.44946657)
 0.327

(c) Since the lower route (A-C-D) has a smaller expected travel time of  TJ = 9 hours as compared to the upper
(with expected travel time = TB = 10 hours), one should take the lower route to minimize expected travel
time from A to D.
Problem 4-2-2
The settlement (in cm) of a structure shown in the following figure may be evaluated from
S = 0.3 A + 0.2 B + 0.1 C
where A, B, and C are respectively the thickness (in m) of the three layers of soil as shown. Suppose A, B, and
C are modeled as independent normal random variables as
A ~ N(5,1)
B ~ N(8,2)
C ~ N(7,1)
(a) Determine the probability that the settlement will exceed 4 cm. (ans. 0.347)
(b) If the total thickness of the three layers is known exactly as 20 m; and furthermore, thicknesses A and B are
correlated with correlation coefficient equal to 0.5, determine the probability that the settlement will exceed
4 cm. (ans. 0.282)
STRUCTURE
A
B
C
Solution:
(a) To calculate probability, we first need to have the PDF of S. As a linear combination of three normal
variables, S itself is normal, with parameters
S = 0.35 + 0.28 + 0.17 = 3.8 (cm)
S = 0.3212  0.2 2 2 2  0.1212  0.51 (cm)
Hence
S  S
4  3.8
)
S
0.509901951
= 1 – (0.39223227) = 1 – 0.65255665
 0.347
P(S > 4cm) = P(

(b) Now that we have a constraint A + B + C = 20m, these variables are no longer all independent, for example,
we have
C = 20 – A – B
Hence
S = 0.3 A + 0.2 B + 0.1(20 – A – B)
 S = 0.2 A + 0.1 B + 2, with AB = 0.5.
Thus
S = 0.25 + 0.18 + 2 = 3.8 (cm) as before, and
0.2 212  0.12 2 2  2  0.5  0.1  1  2
= 0.12  0.346 (cm)
S =
Hence
P(S > 4cm) = 1 – (
4  3 .8
0.12
)
= 1 – (0.57735027) = 1 – 0.718148613  0.282
Problem 4-2-3
A friction pile is driven through three soil layers as shown in the following figure:
40 tons
A
B
C
The total bearing capacity of the friction pile (in tons) is obtained from
Q = 4A + B + 2C
Where A, B, and C are penetration lengths (in meters) through each of the three soil layers, respectively.
Suppose A ~ N(5,3); B ~ N(8,2); A and B are negatively correlated with coefficient  = - 0.5. The total length
of the pile is 30 meters. Determine the probability that the pile will fail to support the 40 ton load, i.e., the event
that the capacity Q is less than 40 tons. [Hint: Observe C = 30 - (A+B)]. (ans. 0.00114)
Solution:
Q = 4A + B + 2C
= 4A + B +2(30 – A – B)
= 2A –B + 60, hence
Q = 25 – 8 + 60 = 62,
Q = [432 + 122 + 2(2)(-1)(-0.5)(3)(2)]1/2
= (36 + 4 + 12)1/2 = 52
Hence P(Q < 40) = P(
Q  Q
Q

40  62
)
52
= (-3.050851079)
 0.00114
Problem 4-2-4
River 1
River 2
City
A city is located at the downstream of the confluence point of two rivers as shown. The annual maximum flood
peak in River-1 has an average of 35 m3/sec with the standard deviation of 10 m3/sec, whereas in River-2 the
mean peak flow rate is 25 m3/sec and the standard deviation is 10 m3 /sec. The annual maximum peak flow rates
in both rivers are normally distributed with a correlation coefficient of 0.5. Presently, the channel which runs
through the city can accommodate up to 100 m 3/sec flow rate without flooding the city. Please answer the
following questions.
(a) What are the mean and standard deviation of the annual maximum peak discharge passing through the city?
(ans. 60 and 17.32 (m3/sec))
(b) What is the annual risk that the city will experience flooding based on the existing channel capacity? What
is the corresponding return period? (ans. 0.0105; 96 years)
(c) Calculate the probability that the city will experience flooding over a 10-year period. (ans. 0.10)
(d) If the desired flooding risk over a 10-year period is to be reduced by half, how large should the present
channel capacity be extended to? (ans. 104.5 m3/sec)
Solution:
(a) Let Q1 and Q2 be the annual maximum flood peak in rivers 1 and 2, respectively. We have
Q1 ~ N(35, 10), Q2 ~ N(25, 10)
The annual max. peak discharge passing through the city, Q, is the sum of them,
Q = Q1 + Q2, hence
 Q   Q1   Q2 = 35 + 25 = 60 (m3/sec), and
 Q2   Q21   Q22 + 2  Q1Q2  Q1  Q2
= 102 + 102 + 20.51010 = 300 (m3/sec),
  Q  300  17.32 (m3/sec)
(b) The annual risk of flooding, p = P(Q > 100)
Q   Q 100  60
= P(
)

Q
300
= 1 – (2.309401077) = 1 – 0.989539359
 0.0105 (probability each year)
Hence the return period is  =
1
p
1
= 95.59643882
0.01046064 1
 96 years.
=
(c) Since the yearly risk of flooding is p = 0.010460641, and we have a course of n = 10 years, we adopt a
binomial model for X, the total number of flood years over a 10-year period.
P(city experiences (any) flooding) = 1 – P(city experiences no flooding at all)
= 1 – P(X = 0)
= 1 – (1 – p)10 = 1 – (1 – 0.010460641)10
= 1 – 0.98953935910 = 1 – 0.900182843
 10%
(d) The requirement on p is, using the flooding probability expression from part (c):
1 – (1 – p)10 = 0.099817157  2 = 0.049908579
 p = 1 – (1 – 0.049908579)1/10 = 0.005106623,
which translates into a condition on the design channel capacity Q0, following what’s done in (b),
Q   Q Q0  60
P(
)=p

Q
300
Q  60
 1 – ( 0
) = 0.005106623
300
Q  60
 ( 0
) = 0.994893377
300
 Q0 = 60 + -1(0.994893377)
300
= 60 + 2.568522177 300 = 104.4881091
Extending the channel capacity to about 104.5 m3/sec will cut the risk by half.
Exercises
Exercise 4-2-1
Fibers may be embedded in cement to increase its strength. Consider a cracked section as shown in the
following diagram:
fibers
crack
T
T
Suppose the total strength T of the cracked section is given by the following expression
T = C + (F1 + F2 + … + FN)
Where C is the strength contributed by the cement; F1, F2, etc. are the strength of each of the fibers across the
crack; N is the total number of fibers across the crack. Suppose C = N(30, 5); each F i is N(5, 3) and N is a
discrete random variable with the following PMF.
PN (n)
0.5
0.3
0.2
0
1
2
n
Assume C and F i ’s are statistically independent. What is the probability that the total strength T will be less
than 30?
Exercise 4-2-2
Footing 1
Footing 2
The settlement of each footing shown follows a normal distribution with a mean of 2 inches and a coefficient of
variation of 30%. Suppose the settlements between two adjacent footings are correlated with a correlation
coefficient of 0.7. Suppose
D = S1 - S 2
where S 1 and S 2 are the settlements of footings 1 and 2, respectively.
(a)
Determine the mean and variance of D.
(b)
What is the probability that the magnitude of the differential settlement (i.e., the difference between the
settlements of two adjacent footings) will be less than 0.5 inch?
Exercise 4-2-3
X3
X1
1
3
4
5
2
X2
The system of pipes shown in the figure is supposed to carry the storm runoffs X 1 and X2, and the municipal
waste water, X3. Suppose the statistics of the flow rates (all in units of cubic feet per second, CFS) are:
X1
X2
X3
Mean
c.o.v.
Distribution
10
15
20
0.30
0.20
0.00
Normal
Normal
----
Because of proximity, X1 and X2 are dependent. Assume that the coefficient of correlation 
X1, X2
= 0.6.
(a)
Determine the mean value and standard deviation of the total rate of inflow to pipe 5.
(b)
What is the probability that during a one minute interval, the volume of water flowing into pipe 2 exceeds
that into pipe 1 by at least 400 cubic feet? (Hint: assume the inflow rate into each pipe is constant during
that minute)
(c)
Suppose the municipal waste water is projected to increase at a rate of 3 cfs per year, and pipe 5 has a
capacity of 70 cfs. If the design criterion is that the probability of overflow at pipe 5 (total inflow exceeds
capacity) after a storm should be less than 0.05, how many years will the current pipe 5 remain adequate,
i.e., before a larger size pipe is needed?
Exercise 4-2-4
The flows in two tributaries 1 and 2 combine to form stream 3 as shown in the following figure.
1
3
2
Stream
Suppose the concentration of pollutants in tributary 1 is X which is normally distributed as N(20, 4) parts per
unit volume (puv); whereas that in tributary 2 is Y which is N(15 ,3) puv. The flow rate in tributary 1 is 600 puv
cubic feet per sec (cfs) and that in 2 is 400 cfs. Hence the concentration of pollutants in the stream is
Z
600 X  400Y
= 0.6X + 0.4Y
600  400
Pollution occurs if pollutant concentration exceeds 20 puv.
(a)
What is the probability that at least one of the two tributaries will be polluted?
(b)
Assume X and Y are statistically independent. Determine the probability of a polluted stream?
(c)
Suppose the same industrial plant dumps the pollutants into the two tributaries; hence X and Y are
correlated, say with ρ= 0.8. What is the probability of a polluted stream?
Exercise 4-2-5
The actual concrete strength, Y, in a structure is generally higher than that, X, measured on specimen from the
same batch of concrete. Data shows that a regression equation for predicting the actual concrete strength is
E( Y x) = 1.12x + 0.05 (ksi);
0.1 < x <0.5
and
Var( Y x) = 0.0025 (ksi)2
Assume that Y follows a normal distribution for a given value of x.
(a)
For a given job where the measured strength is 0.35 ksi, what is the probability that the actual strength
will exceed the requirement of 0.3 ksi?
(b)
Suppose the engineer has lost the data on the measured strength on the concrete specimen. He however
recalls that it is either 0.35 or 0.40 with the relative likelihood of 1 to 4. What is the probability that the
actual strength will exceed the requirement of 0.3 ksi?
(c)
Suppose the measured concrete strength at two sites A and B are 0.35 and 0.4 ksi respectively. What is the
probability that the actual strength for the concrete structure at site A will in fact be larger than that at site
B? you may assume that the predicted actual concrete strength between sites and statistically independent.
Exercise 4-2-6
A pile is designed to have a mean capacity of 20 tons; however, because of uncertainties, the pile capacity is
lognormally distributed with a coefficient of variation (cov) of 20%. Suppose the pile is subject to a maximum
lifetime load that is also lognormally distributed with a mean of 10 tons and a cov of 30%. Assume that pile load
and capacity are statistically independent.
(a)
Determine the probability of failure of the pile.
(b)
A number of piles may be tied together to form a pile group to resist an external load. Suppose the
capacity of the pile group is the sum of the capacities of the individual piles. Consider a pile group that
consists of two single piles as described above. Because of proximity, the capacities between the two piles
are correlated with a correlation coefficient of 0.8. Let T denote the capacity of this pile group.
(i) Determine the mean value and cov of T.
(ii)
Will the random variable T follow a normal, or lognormal or some other probability distribution?
Please substantiate.
Exercise 4-2-7
A contractor submits bids to 3 highway jobs and 2 building jobs. The probability of winning each job is 0.6. The
profit from a highway job is normally distributed as N(100, 40) in thousand dollars; whereas, that for a building
job is N(80, 20) in thousand dollars. Assume that winning each job is an independent event and the profit
between jobs are also statistically independent.
(a)
What is the probability that the contractor will win at least two jobs?
(b)
What is the probability that he will win exactly 1 highway job, but none of the building jobs?
(c)
Suppose he has won 2 highway jobs and 1 building job. What is the probability that the total profit will
exceed 300 thousand dollars?
(d)
How would the answer in part (c) change if the profit form the two highway jobs are correlated with
correlation coefficient ρ= 0.8?
Exercise 4-2-8
During the next month, the amount of water available for a city is lognormally distributed with a mean of 1
million gallons and a c.o.v. of 40%, whereas the total demand is expected to follow a lognormal distribution
with a mean of 1.5 million gallon with a c.o.v. of 10%. What is the probability of a water shortage at this city
over the next month?
Exercise 4-2-9
A corner column of a building is supported on a pile group consisting of 2 piles. The capacity of each pile is a
sum of two independent contributions, namely the frictional resistance, F, along the pile length and the end
bearing resistance, B, at the pile tip. Suppose F and B are both normally distributed with mean values of 20 and
30 tons and coefficients of variation of 0.2 and 0.3, respectively. The capacities between the two piles are
correlated with a correlation coefficient of 0.8.
(a)
Determine the mean and c.o.v. of the capacity of 1 pile.
(b)
Determine the mean and c.o.v. of the total capacity of the pile group.
(c)
If the maximum load applied to the pile group is normally distributed with a mean value of 50 tons and a
c.o.v. of 0.3, what is the probability of failure of this pile group?
Exercise 4-2-10
Consider a tower that is now leaning at an 18° inclination. The tower continues to lean at an annual increment
A, which is normally distributed with a mean value of 0.1° and a c.o.v. of 30%.
(a)
Assume that the incremental amounts of additional lean between years are statistically independent.
Determine the probability that the final inclination of the tower after 16 years will exceed 20°.
(b)
Suppose it is believed that the maximum inclination, M, before the tower collapses (i.e. the leaning
capacity of the tower) is itself a normal random variable with a mean of 20°and a c.o.v. of 1%.
i.
What is the probability that the tower will not collapse within the next 16 years?
ii.
An alternative assumption (instead of statistical independence between years as in part a) is that the tower
will have the same amount of incremental lean each year. What is then the probability that the tower will
not collapse after 16 years?