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Continuous Random Variables: Practice 1 (modified R. Bloom)

Module by: Roberta Bloom. E-mail the authorEdited By: Roberta Bloom

Based on: Continuous Random Variables: Practice 1 by Susan Dean, Barbara Illowsky, Ph.D.

Summary: In this module the student will explore the properties of data with a uniform distribution. The original module of practice problems for the Uniform distribution in Collaborative Statistics by Dr. Barbara Illowsky and Susan Dean has been modified by removing the problems involving conditional probability.

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.

Describe the Data

Exercise 1

What is being measured here?

Solution

The age of cars in the staff parking lot

Exercise 2

In words, define the Random Variable X X size 12{X} {} .

Solution

X X size 12{X} {} = The age (in years) of cars in the staff parking lot

Exercise 3

Are the data discrete or continuous?

Solution

Continuous

Exercise 4

The interval of values for X X size 12{X} {} is:

Solution

0.5 - 9.5

Exercise 5

The distribution for X X size 12{X} {} is:

Solution

X X size 12{X} {} ~ U ( 0 . 5,9 . 5 ) U ( 0 . 5,9 . 5 ) size 12{U \( 0 "." 5,9 "." 5 \) } {}

Probability Distribution

Exercise 6

Write the probability density function.

Solution

f ( x ) f ( x ) size 12{f \( x \) } {} = = 1 9 1 9 size 12{ { {1} over {9} } } {}

Exercise 7

Graph the probability distribution.

  • a. Sketch the graph of the probability distribution.
    Figure 1
    Figure 1 (graph.png)
  • b. Identify the following values:
    • i. Lowest value for X X size 12{X} {} :
    • ii. Highest value for X X size 12{X} {} :
    • iii. Height of the rectangle:
    • iv. Label for x-axis (words):
    • v. Label for y-axis (words):

Solution

  • b.i. 0.5
  • b.ii. 9.5
  • b.iii. Age of Cars
  • b.iv. 1 9 1 9 size 12{ { {1} over {9} } } {}
  • b.v. f ( x ) f ( x ) size 12{f \( x \) } {}

Random Probability

Exercise 8

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.
    Figure 2
    Blank graph with vertical and horizontal axes.
  • b. Find the probability. P ( X < 4 ) P ( X < 4 ) size 12{P \( X<"5730" \) } {} =

Solution

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

Quartiles

Exercise 9

Find the average age of the cars in the lot.

Solution

μ μ size 12{μ} {} = 5

Exercise 10

Find the third quartile of ages of cars in the lot. This means you will have to find the value such that 3 4 3 4 size 12{ { {3} over {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.
    Figure 3
    Blank graph with vertical and horizontal axes.
  • 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"} {} .
  • c. The third quartile is:

Solution

  • b. k k size 12{k} {} = 7.25

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