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 Select one exactly one of 1,2,3,4,5, or 6 the die that is cast the number of times the die was cast the number that appears facing up 0 or 1 a random number between 1 and 6 and is Select one discrete continuous with values Select one all positive real numbers 1, 2, 3, 4, 5, 6 in the interval [1, 6] all real numbers <= 6 0, 1 insufficient information to determine

A die is cast and

X is Select one and is Select one discrete continuous with values Select one  

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


X is Select one 0, 1, 2, 3, ..., or 360 the pointer that was spun the number of times the pointer turned the angle through which the pointer turned 0 through 360 the angle the pointer makes with the vertical and is Select one discrete continuous with values Select one all positive real numbers 1, 2, 3, ..., 360 in the interval [0, 360] all real numbers <= 360 all real numbers in the interval [1, 360]

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

X is Select one and is Select one discrete continuous with values Select one  

The temperature is recorded at midday.

X is Select one the temperature at midday the instrument used the temperature at sunrise the positive real numbers all real numbers the amount of mercury in the thermometer and is Select one discrete continuous with values Select one all positive real numbers all integers between -100 and 212 in the interval [0, 500] all non-negative real numbers all real numbers in the interval [1, 360]
X is Select one and is Select one discrete continuous with values Select one  

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

X is Select one the exact average SAT the rounded average SAT the sample size the number of students in the sample 0, 10, ..., 800 [0, 800] and is Select one discrete continuous with values Select one 0, 10, ..., 800 0, 1, 2, ..., 800 in the interval [0, 800] all non-negative real numbers all real numbers all non-negative integers
X is Select one and is Select one discrete continuous with values Select one  

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.

Applications

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.)

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.
 

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.

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?