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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.

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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.

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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.

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

Now use the distribution to compute the following:

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.

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

Now use the distribution to compute the following:

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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.:

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.:

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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.

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.

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
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