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 Applied calculus exercises: continuous random variables and histograms

In Exercises 1–8, identify the random variable (for example, "X is the price of rutabagas"), decide whether it is continuous or discrete, and choose the most appropriate set of possible values.

A die is cast and the number that appears facing up is recorded.

X is and is with values

A die is cast and

X is and is with values

A dial is spun, and the angle (measured clockwise) the pointer makes with the vertical is noted. (See the figure.)

X is and is with values

A dial of radius one unit is spun (see the figure above), and

X is and is with values

The temperature is recorded at midday.

X is and is with values

X is and is with values

The math SAT of a random sample of college-bound students is rounded to the nearest 10 points, then recorded.

X is and is with values

X is and is with values

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In Exercises 9–12, calculate and adjust the probability distribution histogram of the given continuous random variable by dragging the top of each bar.
Probabilities should be accurate to two decimal places.

Frequency distribution:  \pmb{X} = Height of a jet fighter (thousand feet) 0-20 20-30 30-40 40-50 50-60 Number 10 20 30 40 10

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

Frequency distribution:  \pmb{X} = Time between eruptions of local volcano (thousand years) 0-2 2-3 3-4 4-5 5-6 Frequency

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

Frequency distribution:  \pmb{X} = Average temperature (°F) 0-50 50-60 60-70 70-80 80-90 Number of Cities 4 7 2 5 2

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

Frequency distribution:  \pmb{X} = Cost of a used car (\$1000) 0-2 2-4 4-6 6-8 8-10 Number of cars

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

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### Applications

Farm Population, Female The following table shows the approximate number of females residing on US farms in 1990, broken down by age. Numbers are in thousands.

 Age 0-15 15-25 25-35 35-45 45-55 55-65 65-75 75-95 Number 460 270 250 320 290 290 220 130
Source: Economic Research Service, US Department of Agriculture and Bureau of the Census, US Department of Commerce, 1990.

Construct the associated probability distribution (with probabilities rounded to two decimal places):

 Age 0-15 15-25 25-35 35-45 45-55 55-65 65-75 75-95 Probability

Now use the distribution to compute the following:

 (a) P(15 \leq X \leq 55) = (b) P(X \leq 45) = (c) P(X \geq 45) =

Farm Population, Male The following table shows the approximate number of males residing on US farms in 1990, broken down by age. Numbers are in thousands.

 Age 0-15 15-25 25-35 35-45 45-55 55-65 65-75 75-95 Number
Figures are approximate and randomized. Source: Economic Research Service, US Department of Agriculture and Bureau of the Census, US Department of Commerce, 1990.

Construct the associated probability distribution (with probabilities rounded to two decimal places):

 Age 0-15 15-25 25-35 35-45 45-55 55-65 65-75 75-95 Probability

Now use the distribution to compute the following:

 (a) P(25 \leq X \leq 65) = (b) P(X \leq 15) = (c) P(X \geq 15) =

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Meteors The following partial histogram shows part of the probability distribution of the size (in megatons of released energy) of large meteors that hit the earth's atmosphere. (A large meteor is one that releases at least one megaton of energy, equivalent to the energy released by a small nuclear bomb.)

Based on data released by NASA International Near-Earth-Object Detection Workshop (The New York Times, January 25, 1994, p. C1.)

Calculate or estimate the following probabilities:

 (a) That a large meteor hitting the earth's atmosphere will release between 1 and 4 megatons of energy. (b) That a large meteor hitting the earth's atmosphere will release between 3 and 4.5 megatons of energy. (c) That a large meteor will release at least 5 megatons of energy.

Meteors The following partial histogram shows part of the probability distribution of the size (in megatons of released energy) of large meteors that hit the atmosphere of planet Zor in the Cygnus III system in Andromeda.

Based on data released by the Imperial Zor Near-Planet-Object Detection Workshop (The Zor Chronicle, Saturno 25, 6994, p. C1.)

Calculate or estimate the following probabilities, rounded to three decimal places:

 (a) That a large meteor hitting Zor's atmosphere will release between and megatons of energy. (b) That a large meteor hitting Zor's atmosphere will release between and megatons of energy. (c) That a large meteor hitting Zor's atmosphere will release megatons of energy.

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Quality Control An automobile parts manufacturer makes heavy-duty axles with a cross-section radius of 2.3 cm. In order for one of its axles to meet the accuracy standard demanded by the customer, the radius of the cross section cannot be off by more than 0.02 cm. Construct a probability distrbution histogram with X = the measured radius of an axle, using categories of width 0.01 cm, so that all of the following conditions are met.

(a) X is always between 2.26 and 2.34.
(b) 80% of the axles have a cross-section radius between 2.29 and 2.31.
(c) 10% of the axles are rejected.

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

Damage Control As a campaign manager for a presidential candidate who always seems to be getting himself into embarrassing situations, you have decided to conduct a statistical analysis of the number of times per week he makes a blunder. Construct a probability histogram with X = the number of times he blunders in a week, using categories of width 1 unit, so that all of the following conditions are met.

(a) X is always between and (inclusive).
(b) During a given week, there is an 80% chance that he will make to blunders.
(c) Never a week goes by when he doesn't make at least blunders.
(d) On occasion, he has made blunders in one week.

Probability distribution histogram: To adjust the histogram drag each bar from its top edge to the appropriate height.

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### Communication and Reasoning Exercises

How is a random variable related to the outcomes in an experiment?

Give an example of an experiment and two associated continuous random variables.

You are given a probability distribution histogram with the bars having a width of 2 units. How is the probability P(a \leq X \leq b) related to the area of the corresponding portion of the histogram?

You are given a probability distribution histogram with the bars having a width of 1 unit, and you wish to convert it into one with bars of width 2 units. How would you go about this?

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Last Updated: February, 2008