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Continuous Random Variables: Practice 1

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: In this module the student will explore the properties of data with a uniform distribution. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

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Student Learning Outcomes

The student will explore the properties of data with a uniform distribution

Given

The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

Properties of the Data

Exercise 1

  1. What is being measured here?
  2. In words, define the Random Variable X X size 12{X} {} . X X size 12{X} {} =______________________________.
  3. Are the data continuous or discrete?
  4. The interval of values for X X size 12{X} {} is:______________________
  5. X X size 12{X} {} ~________________________

Solution

  1. Age of cars in staff lot.
  2. Age of cars in staff lot.
  3. Continuous
  4. 0.5 - 9.5
  5. U ( 0 . 5,9 . 5 ) U ( 0 . 5,9 . 5 ) size 12{U \( 0 "." 5,9 "." 5 \) } {}

Probability Distribution

Exercise 2

  1. Write the probability density function: f ( x ) f ( x ) size 12{f \( x \) } {} :_________________________________
  2. Sketch the graph of the probability distribution. Include:
    • lowest value for X X size 12{X} {} =______________________ highest value for X X size 12{X} {} =______________________
    • labeling on xx size 12{x} {}-axis (words): _____________________________________
    • height of rectangle = __________ labeling on yy size 12{y} {}-axis: __________
    • graph.png

Solution

  1. 1 9 1 9 size 12{ { {1} over {9} } } {}
  2. 0.5; 9.5; age of cars; 1 9 1 9 size 12{ { {1} over {9} } } {} ; f ( x ) f ( x ) size 12{f \( x \) } {}

Random Probability

Exercise 3

Find the probability that a randomly chosen car in the lot was less than 4 years old.

    • a.: Sketch the graph. Shade the area of interest.
    • graph.png
    • b.: Find the probability. P ( X < 4 ) P ( X < 4 ) size 12{P \( X<4 \) } {} :___________________________
  2. Out of just the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than 4 years old.
    • a.: Sketch the graph. Shade the area of interest.
    • graph.png
    • b.: Find the probability. P ( X < 4 ∣ X < 7 . 5 ) P ( X < 4 ∣ X < 7 . 5 ) size 12{P \( X<4 \lline X<7 "." 5 \) } {} =______________________
  3. Discussion question: What has changed in problems (1) and (2) above to make the solutions different?

Solution

    • b.: 3 . 5 9 3 . 5 9 size 12{ { {3 "." 5} over {9} } } {}
    • b.: 3 . 5 7 3 . 5 7 size 12{ { {3 "." 5} over {7} } } {}

Quartiles

Exercise 4

  1. Find the average age of cars in the lot. μ μ size 12{μ} {} :_________________________
  2. Find the third quartile of ages of cars in the lot. This means you will have to find the value such that 3/4, or 75%, of the cars are at most (less than or equal to) that age.
    • a.: Sketch the graph. Shade the area of interest.
    • graph.png
    • b.: Find the value k k size 12{k} {} such that P ( X < k ) = 0 . 75 P ( X < k ) = 0 . 75 size 12{P \( X<k \) =0 "." "75"} {} . k k size 12{k} {} =_________________________ Recall that you are looking for a critical value, not a probability. The probability is given to be 0.75.)
    • The third quartile is ____________________

Solution

  1. 5
    • b.: 7.25

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