Engineering Optimization: Theory and Practice, Fourth Edition

Problems 663

11.5 What is a random variable?

11.6 Give two examples of random design parameters.

11.7 What is the difference between probability density and probability distribution functions?

11.8 What is the difference between discrete and continuous random variables?

11.9 How does correlation coefficient relate two random variables?

11.10 Identify possible random variables in a LP problem.

11.11 How do you find the mean and standard deviation of a sum of several random variables?

Problems

11.1 A contractor plans to use four tractors to work on a project in a remote area. The

probability of a tractor functioning for a year without a break-down is known to be

80 %. IfXdenotes the number of tractors operating at the end of a year, determine the

probability mass and distribution functions ofX.

11.2 The absolute value of the velocity of a molecule in a perfect gas (V )obeys the Maxwell

distribution

fV(ν)=

4 h^3

√

π

ν^2 e−h

(^2) ν 2
, ν≥ 0
whereh^2 =(m/ 2 kT )is a constant (mis the mass of the molecule,kis Boltzmann’s
constant, andTis the absolute temperature). Find the mean and the standard deviation
of the velocity of a molecule.
11.3 Find the expected value and the standard deviation of the number of tractors operating
at the end of one year in Problem 11.1.
11.4 Mass-produced items always show random variation in their dimensions due to small
unpredictable and uncontrollable disturbing influences. Suppose that the diameter,X, of
the bolts manufactured in a production shop follow the distribution
fX(x)=a(x− 0. 9 )( 1. 1 −x) for 0. 9 ≤x≤ 1. 1
0 elsewhere
Find the values ofa,μXandσX^2.
11.5 (a)The voltageVacross a constant resistanceRis known to fluctuate between 0 and
2 volts. IfV follows uniform distribution, what is the distribution of the power
expended in the resistance?
(b)Find the distribution of the instantaneous voltage (V )given byV=Acos(ωt+φ),
whereAis a constant,ωthe frequency,tthe time, andφthe random phase angle
uniformly distributed from 0 to 2πradians.
11.6 The hydraulic head loss (H )in a pipe due to friction is given by the Darcy–Weisbach
equation,
H=f
L
2 gD
V^2
wherefis the friction factor,Lthe length of pipe,Vthe velocity of flow in pipe,gthe
acceleration due to gravity, andDthe diameter of the pipe. IfVfollows exponential