Applied Statistics and Probability for Engineers

5-15

Thus, the probability that is within 3(0.01^2 n)^1 ^2 of is at least 89. Finally, nis chosen

such that 3(0.01^2 n)^1 ^2 0.005. That is,

EXERCISES FOR SECTION 5-10

n 323 0.01^2 0.005^24  36

X

S5-25. The photoresist thickness in semiconductor manu-

facturing has a mean of 10 micrometers and a standard devia-

tion of 1 micrometer. Bound the probability that the thickness

is less than 6 or greater than 14 micrometers.

S5-26. Suppose Xhas a continuous uniform distribution

with range 0 x10. Use Chebyshev’s rule to bound the

probability that Xdiffers from its mean by more than two stan-

dard deviations and compare to the actual probability.

S5-27. Suppose Xhas an exponential distribution with

mean 20. Use Chebyshev’s rule to bound the probability that

Xdiffers from its mean by more than two standard deviations

and by more than three standard deviations and compare to the

actual probabilities.

S5-28. Suppose Xhas a Poisson distribution with mean 4.

Use Chebyshev’s rule to bound the probability that Xdiffers from

its mean by more than two standard deviations and by more than

three standard deviations and compare to the actual probabilities.

S5-29. Consider the process of drilling holes in printed cir-

cuits boards. Assume that the standard deviation of the diame-

ters is 0.01 and that the diameters are independent. Suppose

that the average of 500 diameters is used to estimate the

process mean.

(a) The probability is at least 1516 that the measured aver-

age is within some bound of the process mean. What is the

bound?

(b) If it is assumed that the diameters are normally distrib-

uted, determine the bound such that the probability is

15 16 that the measured average is closer to the process

mean than the bound.

S5-30. Prove Chebyshev’s rule from the following steps.

Define the random variable Yas follows:

(a) Determine E(Y)

(b) Show that

(c) Using part (b), show that

(d) Using part (c), complete the derivation of Chebyshev’s

inequality.

E 31 X 224 c^2   2 E 3 Y 4

1 X 22  1 X 22 Yc^2   2 Y

Ye

1 if 0 X 0 c

0 otherwise

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