Chapter 7.2 B: Rules for Means and Variances
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1. Suppose 𝜇𝑥 = 5 and 𝜇𝑦 = 10. According to the rules for means, what is 𝜇𝑥+𝑦 ? 15
2. Suppose 𝜇𝑥 = 2. According to the rules for means, what is 𝜇3+4𝑥 ? 3+4(2) = 11
3. Suppose 𝜎𝑥2 = 2 and 𝜎𝑦2 = 3 and X and Y are independent random variables. According to the rules for variances, what is ?
2
What is 𝜎𝑥+𝑦
? What is 𝜎𝑥+𝑦 ? 5
2
4. Suppose 𝜎𝑥2 = 4. According to the rules for variances, what is 𝜎3+4𝑥
? What is 𝜎3+4𝑥 ? 64, and 8
5. What is the best way to combine standard deviations? Square the standard deviations and add the variances then take the
square root.
6. Consider the following distribution for some random variable X:
X:
1
2
5
P(X):
0.2
0.5
0.3
a)
Let a = 2 and b = 3. Determine the probability distribution function for the new random variable 𝑎 + 𝑏𝑋.
X:
5
8
17
P(X):
0.2
0.5
0.3
b)
Use the definitions (pages 483 and 485) to find the mean and variance of 𝑎 + 𝑏𝑋.
c)
Now use Rule 1 for means (page 494) to find the mean and variance of 𝑎 + 𝑏𝑋. Verify that your results in b) and c) are
the same.
d)
Verify that Rule 1 for variances holds for this example. That is, show that 𝑣𝑎𝑟(2 + 3𝑥) = 9𝑣𝑎𝑟(𝑋)
e)
Which method do you prefer: using the definition or using the rules? Why?
7. Checking for independence. For each of the following situations, would you expect the random variables X and Y to be
independent? Explain your answers.
a) X is the rainfall (in inches) on November 6 of this year, and Y is the rainfall at the same location on November 6 of the next
year.
b) X is the amount of rainfall today, and Y is the amount of rainfall at the same location tomorrow.
c) X is today’s rainfall at the airport in Orlando, Florida, and Y is today’s rainfall in Disney World just outside Orlando.
8. Checking for independents. In which of the following games of chance would you be willing to assume independence of X
and Y in making a probability model? Explain your answer in each case.
a) In blackjack, you are dealt two cards and examine the total points X on the cards (face cards count 10 points). You can
choose to be dealt another card and compete based on total points Y on all three cards.
b) In craps, the betting is based on successive rolls of two dice. X is the sum of the faces on the first roll, and Y is the sum of the
faces on the next roll.
9. Laboratory data show that the time required to complete two chemical reactions in a production process varies. The first
reaction has a mean time of 40 minutes and a standard deviation of 2 minutes, the second has a mean of 25 minutes and a
standard deviation of 1 minute. The two reactions are run in sequence during a production. There is a fixed period of 5 minutes
between them as the product of the first reaction is pumped into the vessel where the second reaction will take place. What is the
mean time required for the entire process?
70 minutes
10. A time and motion study measures the time required for an assembly-line worker to perform a repetitive task. The data show
that the time required to bring a part from a bin to its position on an automobile chassis varies from car to car with mean 11
seconds and standard deviation 2 seconds. The time required to attach the part to the chassis varies with mean 20 seconds and
standard deviation 4 seconds.
a) What is the mean time required for the entire operation of positioning and attaching the part?
b) If the variation in the worker’s performance is reduced by better training, the standard deviation will decrease. Will this
decrease change the mean you found in part a) if the mean time for the two steps remains the same?
c) The study finds that the times required for the two steps are independent. A part that takes a long time to position, for
example, does not take more or less time to attach than the other parts. Find the standard deviation of the time required for the
two-step assembly operation.
11. The design of an electrical circuit calls for a 100-ohm resistor and a 250-ohm resistor connected in a series so that their
resistances add. The components used are perfectly uniform, so that the actual resistances vary independently according to
Normal distributions. The resistance of 100-ohm resistors has mean 100 ohms and standard deviation 2.5 ohms, while that of
250-ohm resistors has mean 250 ohms and standard deviation 2.8 ohms.
a) What is the distribution of the total resistance of the two components in seris?
b) What is the probability that the total resistance lies between 345 and 355 ohms? Show your work, and include drawings of
the interval you are interested in the Normal curve describing the two random variables x and y combined.