Calculus Applied to Probability and Statistics
by
Stefan Waner and Steven R. Costenoble

Exercises
for
Section 1: Continuous Random Variables and Histograms

1. Continuous Random Variables and Histograms

2. Probability Density Functions: Uniform, Exponential, Normal, and Beta

Calculus and Probability Main Page

"Real World" Page

Answers to Odd-Numbered Exercises

In Exercises 1 through 10, identify the random variable (for example, "X is the price of rutabagas"), say whether it is continuous or discrete, and if continuous, give its interval of possible values.

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

2. A die is cast and the time it takes for the die to become still is recorded.

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



4. A dial is spun, and the quadrant in which the pointer comes to rest is noted.

5. The temperature is recorded at midday.

6. The U.S. Balance of Payments is recorded (fractions of a dollar permitted).

7. The U.S. Balance of Payments is recorded, rounded to the nearest billion dollars.

8. The time it takes a new company to become profitable is recorded.

9. In each batch of 100 computer chips manufactured, the number that fail to work is recorded.

10. The time it takes a TV set to break down after sale is recorded.

In Exercises 11-14, sketch the probability distribution histogram of the given continuous random variable.

11.

X = height of a jet fighter (ft.)	0-20,000	20,000-30,000	30,000-40,000	40,000-50,000	50,000-60,000
Probability	0.1	0.2	0.3	0.3	0.1

12.

X = time to next eruption of a volcano (yrs.)	0-2,000	2,000-3,000	3,000-4,000	4,000-5,000	5,000-6,000
Probability	0.1	0.3	0.3	0.2	0.1

13.

X = average temperature (šF)	0-50	50-60	60-70	70-80	80-90
Number of Cities	4	7	2	5	2

14.

X = Cost of a used car ($)	0-2,000	2,000-4,000	4,000-6,000	6,000-8,000	8,000-10,000
Number of Cars	200	500	800	500	500

Applications

15. Farm Population, Female The following table shows the 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	459	265	247	319	291	291	212	126

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 four decimal places) and use the distribution to compute the following.

(a) P(15 X 55)
(b) P(X 45)
(c) P(X 45)

16. Farm Population, Male The following table shows the 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	480	324	285	314	302	314	247	118

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 four decimal places) and use the distribution to compute the following.

(a) P(25 X 65)
(b) P(X 15)
(c) P(X 15)

17. Meteors The following 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.)

The authors' model, 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.

18. Meteors Repeat the preceeding exercise using the following histogram for meteor impacts on the planet Zor in the Cygnus III system in Andromeda.

19. 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 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 lies in the interval [2.26, 2.34].
(b) 80% of the axles have a cross-sectional radius between 2.29 and 2.31.
(c) 10% of the axles are rejected.

20. 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 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 lies in the interval [0, 10].
(b) During a given week, there is an 80% chance that he will make 3 to 5 blunders.
(c) Never a week goes by when he doesn't make at least one blunder.
(d) On occasion, he has made 10 blunders in one week.

Communication and Reasoning Exercises

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

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

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

24. 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?

1. Continuous Random Variables and Histograms

2. Probability Density Functions: Uniform, Exponential, Normal, and Beta

Calculus and Probability Main Page

"Real World" Page

Answers to Odd-Numbered Exercises

We would welcome comments and suggestions for improving this resource. Mail us at:

Stefan Waner (matszw@hofstra.edu)

Steven R. Costenoble (matsrc@hofstra.edu)

Last Updated: September, 1996 Copyright © 1996 StefanWaner and Steven R. Costenoble