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5-15

2
Thus, the probability that X is within 3(0.01 n) 1 2 of is at least 8 9. Finally, n is chosen
2
such that 3(0.01 n) 1 2 0.005. That is,
2
2
2
n 3 30.01 0.005 4 36
EXERCISES FOR SECTION 5-10
S5-25. The photoresist thickness in semiconductor manu- that the average of 500 diameters is used to estimate the
facturing has a mean of 10 micrometers and a standard devia- process mean.
tion of 1 micrometer. Bound the probability that the thickness (a) The probability is at least 15 16 that the measured aver-
is less than 6 or greater than 14 micrometers. age is within some bound of the process mean. What is the
S5-26. Suppose X has a continuous uniform distribution bound?
with range 0 x 10. Use Chebyshev’s rule to bound the (b) If it is assumed that the diameters are normally distrib-
probability that X differs from its mean by more than two stan- uted, determine the bound such that the probability is
dard deviations and compare to the actual probability. 15 16 that the measured average is closer to the process
mean than the bound.
S5-27. Suppose X has an exponential distribution with
mean 20. Use Chebyshev’s rule to bound the probability that S5-30. Prove Chebyshev’s rule from the following steps.
X differs from its mean by more than two standard deviations Define the random variable Y as follows:
and by more than three standard deviations and compare to the
actual probabilities. 1 if 0X 0 c
Y e
S5-28. Suppose X has a Poisson distribution with mean 4. 0 otherwise
Use Chebyshev’s rule to bound the probability that X differs from
its mean by more than two standard deviations and by more than (a) Determine E(Y)
three standard deviations and compare to the actual probabilities. (b) Show that 1X 2 1X 2 Y c Y
2
2 2
2
2 2
2
S5-29. Consider the process of drilling holes in printed cir- (c) Using part (b), show that E31X 2 4 c E 3Y4
cuits boards. Assume that the standard deviation of the diame- (d) Using part (c), complete the derivation of Chebyshev’s
ters is 0.01 and that the diameters are independent. Suppose inequality.