Download ISE 261 HOMEWORK THREE Due Date: Thursday 2/26/2015 1

ISE 261
HOMEWORK THREE
Due Date: Thursday 2/26/2015
1. Computer chips often contain surface imperfections. For a certain type of computer chip, 9%
contain no imperfections, 23% contain 1 imperfection, 27% contain 2 imperfections, 19%
contain 3 imperfections, 13% contain 4 imperfections, and the remaining 9% contain 5
imperfections. Let X represent the number of imperfections in a randomly chosen chip. Is X
discrete or continuous? What is the probability distribution for X?
Ans:
2. The number of flaws in a 1-length of copper wire produced by a manufacturing process
varies from wire to wire. Overall, 46% of the wires produced have no flaws, 38% have one
flaw, 12% have two flaws, and 4% have three flaws. Let X represent the number of flaws in a
randomly selected piece of wire and let F(x) denote the cumulative distribution function. Find
F(2). Find F(1.5).
Ans:
3. A certain industrial process is brought down for recalibration whenever the quality of the
items produced falls below specifications. Let X represent the number of times the process is
recalibrated during a week, and assume X has the probability mass function shown below. Find
the mean of X.
X
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|
0
P(X) |
0.39
1
2
3
4__
5
0.23 0.18 0.15 0.04 0.01
4. Find the variance for the random variable X described in problem #3. (Use short cut).
Ans:
5. A resistor in a certain circuit is specified to have a resistance in the range 99 Ω-101 Ω. An EE
obtains two resistors. The probability that both of them meet the specification is 0.40, and the
probability that exactly one of them meets the specification is 0.50, and the probability that
neither of them meets the specification is 0.10. Let random variable X represent the number
resistors that meet the specification. Find the probability mass function and mean of X.
Ans:
6. Benzene (C6H6 – the simplest aromatic hydrocarbon), commonly used to synthesize plastics
in industry, was discovered by Michael Faraday in 1825 in illuminating gas made from whale
oil. The hydrogentation of benzene to cyclohexane is promoted with a finely divided porous
nickel catalyst. The catalyst particles can be considered to be spheres of various sizes. Assume
all the particles have integer masses between 10 and 30 μg. Let X be the mass of a randomly
chosen particle. The probability mass function of X is given by the formula shown below. What
proportion of particles has masses less than 17 μg? Also, provide the mean and standard
deviation of the particle masses.
p(X) = (x – 10) / 190 for x = 11, 12, 13, …, 29
Ans:
0 other wise
7. A small market orders copies of a certain magazine for its magazine rack each week. Let
random variable X equal to demand for the magazine, with a probability mass function shown
below. Suppose the store owner actually pays $2.00 for each copy of the magazine and the price
to customers is $3.00. If magazines left at the end of the week have no salvage value, is it better
to order three or four copies of the magazine? [Hint: For both three and four copies ordered,
express net revenue as a function of demand X, and then compute the expected revenue.]
x
|
P(x) |
Ans:
1
2
3
4
5
6___ 7__
1/16 2/16 3/16 4/16 3/16 1/16
8
1/16 1/16
8. Nearly all-commercial industrial facilities have three-phase power systems, especially for
motor and heavy machinery loads (see diagram below). With three-phase power systems comes
the expectation that if the three-phases have about the same amount of current, then the neutral
will have very little current, thus electrical engineers try to balance the loads so equal amounts
of current are drawn on each of the three power legs. There are a number of reasons why such
load balancing is desirable, but one reason is that it often results in the least current flowing in
the neutral conductor. If high currents are measured in the neutral, or at least higher than we
expect, problems may result (e.g. overheating & fire). Also, electrical engineers recognize that
high neutral current in computer power systems is a potential problem. A survey of computer
power system load currents at US sites found that 20% of the sites had high neutral to full-load
current ratios. If a random sample of seven computer power systems is selected from the large
number of sites in the country, what is the probability that at least five will have a high neutral
to full-current load ratio?
Ans:
9. A prominent physician claims that 90% of those with lung cancer are chain smokers. If his
assertion is correct, find the probability that of eight such patients recently admitted to a
hospital, fewer than half are chain smokers.
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10. A family decides to have children until it has three children of the same gender. Assuming
P(B) = P(G) = 0.5, what is the pmf of random variable X = the number of children in the
family?
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11. A taxi company has a small limousine at the local airport with a seating capacity of four that
makes one trip per day to a predefined location. The cost of each seat to the taxi company is
estimated at $5 per seat (empty seats still cost $5). The price per seat to the customer is $10.
The number of reservation requests per day X is Poisson distributed with parameter λ = 4 for x
= 0,1,2... Assuming any reservation requests over four will go to rival limousine services (the
reservation request becomes lost to this taxi company), find the expected net revenue. (Note: the
limousine driver is still required to make the trip even if there are no customers to be seated).
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12. What is the probability of observing 5 red cards in 5 draws from an ordinary deck of 52
playing cards?
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13. An electronic scale in an automated filling operation stops the manufacturing line after five
underweight packages are detected. If the probability of an underweight package is 0.002 and
fill attempts are independent, how many fills would you expect this line to experience before
the line is stopped?
Ans:
T
TH
HE
EE
EN
ND
D