1. Linear algebra and probability
In the fabrication of semiconductors, an industrial manufacturer has a process that produces
silicon wafers. Sometimes, the wafers come out defective, and have to be discarded.
(a) Every wafer has a production cost c. Every correct wafer can be sold for v; defective wafers
cannot be sold and bring in no money. Assuming the probability of a defective wafer is
written P(D = 1), where D is a random variable that may take on the values 0 and 1, write
down the expression for the expected proﬁt of manufacturing a single wafer. 
(b) The defect rate in one production line is 1:100000. The company learns of a new device that
can predict whether the wafer will be defective after only a few seconds with a reliability of
99%. This will allow the fabricator to pull the wafer and abort the production process before
a more expensive process begins. Explain how Bayes’ rule would help the fabricator decide
if this device is a worthwhile investment and give your recommendation based on these
ﬁgures. You do not have to provide an exact calculation but you should give approximate
ﬁgures. State any assumptions you make.
(c) The production system is to be optimized to minimize the number of defects. The fabrication
process has two adjustable parameters: temperature and slice thickness (that have been
normalised so they are on a unit-less scale).
(i) Slice thickness (st) and temperature (temp) have been set every day at noon over ten
days. The raw data:
Make a sketch of a scatter plot of scale thickness (”x-axis”) vs temperature (”y-axis”)
and include an indication of the two principle components. Deﬁne what principal
components are and how they can be computed.
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