Mathematical and Computational Methods for Engineers
E155C, Winter 2004
Problem Set #3
(Discrete and Continuous Distributions, Distributions of Several Random Variables)
Date: 1/21/2004
Due: 1/28/2004
50% of the citizens of this country have a
below average understanding of statistics.
Reading:
Binomial distribution: Ross 5.1
Poisson distribution: Ross 5.2
Uniform distribution: Ross 5.4
Normal distribution: Ross 5.5
Jointly distributed random variables: Ross 4.3
Covariance, correlation coefficient: Ross 4.7
MATLAB Workbook: Exercise 2
Problem 1 Experience shows that 10% of the individuals reserving tables at a nightclub
will not appear. If the night club has 50 tables and takes 53 reservations, what is the
probability that it will be able to accommodate everyone appearing ?
Problem 2 Electrical power failures in a workplace have been modeled as a Poisson
experiment with a rate of 1.5 per month
a) what is the probability of having more than 20 failures in a year ?
b) what is the probability that the number of failures in a year will differ by more
than a standard deviation from the expected number ?
Problem 3 It was discussed in class that the Poisson distribution:
(t ) x e  t
f ( x) 
x!
can be used to model the probability that exactly X events occur in any given time
interval of length t, where  is the arrival rate. Use the Poisson distribution to show that
the interarrival time (i.e. the time interval between any two consecutive events) possesses
the exponential distribution:
f (t )  et
t0
Problem 4 The exponential density function in the previous problem:
f (t )  et
t0
is often used to model the failure of equipment components. That is, the probability that a
particular component will fail within time T is given by:
T
Pf   f (t )dt
0
Reliability is defined as the probability that a component will not fail in time T, i.e.
R  1  Pf
a) what is the expected (average) lifetime of a component in terms of  ?
b) if an electronic component has the average lifetime of 10,000 hrs, for what time
span might we expect its reliability to be 0.99 ?
Problem 5 It is generally known that redundancy at the component level is more
effective than redundancy at the system level in improving the overall system reliability
using the same number of components. Consider two alternative configurations
consisting of three redundant components A, B, and C as shown below. Assume that all
components have the same reliability of 0.99. In both cases, find the overall system
reliability and make a recommendation as to which of the two approaches should be used.
A
B
C
A
B
C
V
A
B
C
A
B
C
Problem 6 When requirements for high reliability make redundancy necessary, a good
arrangement has one unit operating until it fails. At that time, a second unit, which has
been idly standing by is switched on. The overall reliability of such a system, typically
referred to as a standby system, is generally higher than the reliability of an equivalent
parallel system in which both units operate simultaneously. A two-unit standby system
functions successfully when the functioning unit does not fail, or if the functioning unit
fails during operating time t, the sensing/switching unit functions properly, and the
standby unit (not having failed while idle) functions properly for the remainder of the
mission. Assuming 100% reliabilities of the sensing/switching unit and the second unit
while idling, the reliability of the system is the probability that unit 1 succeeds for the
whole period t or that unit 1 fails at some time t1 prior to t and unit 2 successfully
functions for the remainder of the mission. If the failure rates for the first and second
units are 1 and 2 , respectively, determine the system reliability as a function of time for
the following two cases: 1  2 and 1  2 . For the case where 1  2 , determine the
mean time to failure and compare your result with the mean time to failure of a parallel
system with two redundant components. Comment on the results.
Problem 7 In producing ball bearings, the manufacturing process has shown a radius
variation of  2  0.09 mm2. Using Chebyshev’s inequality, it was shown in class that the
upper bound on the probability that the radius falls more than 0.9 mm away from the
mean was 0.11 (see Example 5 in Handout #2). Suppose now that the radius is normally
distributed. Determine the actual probability that the radius will fall 0.9 mm away from
the mean and compare your result to the upper bound.
Problem 8 The tolerance limits for a circuit breaker are 40  0.5 Amperes. This means
that any circuit breaker that breaks at an amperage less than 39.5 breaks at too low a level
and any that breaks at an amperage greater than 40.5 breaks at too high a level. If a
shipment of circuit breakers possesses break points that are normally distributed with
mean 39.3 and standard deviation 0.2, then what percentage of the shipment is defective ?
(i.e. outside the specified tolerance limits)
Problem 9 Flight time of an air carrier is normally distributed with mean 2.34 hours and
standard deviation 0.28 hours. If it is desired that the flight arrive at its destination on
time 95% of the time, what flying time should be allowed ?
Problem 10 Let X and Y be independent random variables with probability density
functions fx(x) and fy(y) respectively.
a) show that the density for Z = X + Y is given by:

f Z ( z) 
f
X
( x) fY ( z  x)dx

[Hint: consider an incremental perturbation from z to z+dz and find
P( z  Z  z  dz ) ]
b) use your result in part a) to verify that the characteristic function for the sum of
two independent random variables is equal to the product of the corresponding
characteristic functions:
 Z (w)   X (w) Y (w)
[Hint: this property was already derived in lecture in a different way]
Problem 11 Let f ( x, y )  k , if x  0, y  0 , and x  y  3 and 0 otherwise.
a) find k
b) find P( X  Y  1)
c) find P( X 2  Y 2  1)
d) find P(Y  X )
e) determine whether or not X and Y are independent
Problem 12 Monte Carlo analysis is a mathematical simulation of physical processes
based on one’s knowledge of the distributions of the process parameters. Monte Carlo
simulations are used to predict the performance of complex physical systems that are
subject to uncertainty and whose outcomes are controlled by the interaction of multiple
independent factors.
A three-component system, shown below, consists of a top plate, a base plate, and a bolt
that must fit through holes in the two plates simultaneously. The top plate is not free to
move relative to the base plate. Since there is variation in the diameters of the holes and
bolt and in the true positions of the holes due to manufacturing and/or assembly
tolerances, there will be some chance that the alignment will be off enough that the bolt
will not fit through the two holes.
Monte Carlo analysis can be used in this situation to determine the probability of an
arbitrary set of components fitting together and evaluating the tolerance specifications on
the individual components and the alignment process.
a) If the radius of the hole in the top plate is r p , the
radius of the hole in the base plate is rb , the bolt
diameter is d , and the relative misalignment of
the two holes is r , determine the geometric
criterion that can be used to determine whether
the bolt will or will not fit into the assembly.
Consider the following two cases: r  rp  rb
rb
r
and r  rp  rb , as the “fitting” criterion is
rp
different for the two cases. Refer to the diagram
for details.
b) The table below contains the nominal dimensions and 3 tolerance limits for the
radii of the two holes, the diameter of the bolt, and the x and y coordinates of the
two holes measured with respect to some reference point. It is assumed that all the
dimensions and coordinates are normally distributed.
Item
Top plate hole radius
Base plate hole radius
Bolt radius
Top plate hole x-position
Top plate hole y-position
Base plate hole x-position
Base plate hole y-position
Nominal value
(mils)
252.5
251.5
249.5
1000
1000
1000
1000
Tolerance
(mils)
 0.5
 0.5
 2.1
 2.0
 2.0
 2.0
 2.0
Write a MATLAB script to simulate the outcomes of 1000000 trials. Determine the
relative proportion of the outcomes in which the bolt will not fit into the assembly.
Comment on whether or not the resultant probability is adequately low and how you
would go about increasing/decreasing the odds of having defective parts. Note that
maintaining very low probabilities of finding defective parts is not necessarily optimal as
it implies very tight tolerances and, therefore, high manufacturing and assembly costs.
[Hint: to generate a vector of samples drawn from the normal distribution use the
normrnd function. Type help normrnd for syntax information. Note that it is
computationally more efficient to generate all 1000000 values for each of the variables at
once as a vector, rather than drawing samples one by one]