Math Answers

9. The probability density function for a diameter of a drilled hole in millimeters is f(x) =

10e^−10(x−5)

for x > 5 mm. Although the target diameter is 5 mm, vibrations, tool wear,

and other nuisances produce diameters larger than 5 mm.

a) Determine the mean and variance of the diameter of the holes. [Hint: Use

integration by parts.]

b) Determine the probability that the hole exceeds 5.1 mm.

8. Suppose the probability density function of the length of computer cables is f(x) = 0.1

from 1200 to 1210 millimeters.

a) Determine the mean and standard deviation of the cable length.

b) If the length specifications are 1195 < x < 1205, what proportions of cables are

within specifications?

The density function of the thickness of a conducive coating in micrometers is f(x) =

600x^−2

for 100 < x < 120.

a) Determine the mean and variance of the coating thickness.

b) If the coating costs $0.50 per micrometer of thickness on each part, what is the

average cost of the coating per part?

Suppose that contamination particle size (in micrometers) can be modeled as f(x) =

2x^−3

for x > 1. Determine the mean and standard deviation of X.

5. The maximum patent life for a new drug is 17 years. Subtracting the length of time

required by the FDA for testing and approval of the drug provides the actual patent life

of the drug – that is, the length of time that a company has to recover research and

development costs and make profit. Suppose the distribution of the lengths of patent life

for new drugs is as shown here:

x 3 4 5 6 7 8 9 10 11 12 13

f(x) 0.03 0.05 0.07 0.10 0.14 0.20 0.18 0.12 0.07 0.03 0.01

a) Find the expected number of years of patent life for a new drug.

b) Find the standard deviation of x.

c) Find the probability that x falls into the interval μ ± 2σ.

4. You can insure a $50,000 diamond for its total value by paying a premium of D dollars. If

the probability of theft in a given year is estimated to be 0.01, what premium should the

insurance company charge if it wants the expected gain to equal $1000?

3. Suppose that X has a discrete uniform distribution on the integers 0 through 9.

Determine the mean, variance, and standard deviation of the random variable Y = 5X

and compare to the corresponding results for X.

Seven students went on a diet in an attempt to lose weight, with two of them losing weight while all the others added weight. Is the diet is an effective way to losing weight α=1% (5mks)

2. A fire-detection device uses three-temperature-sensitive cells acting independently of

one another in such a manner that any one or more can activate the alarm. Each cell has

a probability p = 0.8 of activating the alarm when the temperature reaches 100° or

higher. Let X represent the number of cells activating the alarm when the temperature

reaches 100°. Find:

a) the probability distribution of X;

b) the expected value; and

c) the variance for the random variable X.

A random sample of size 36 was taken from a population distributed as Nμ,3.92.The value of the sample x was 15.6. i. Find a 90% confidence interval for μ. (5mks)

It is believed that value of μ is 17. Use your confidence interval to comment on this belief.(2mks)