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Continuous Random Variables: Practice 1

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

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Summary: In this module the student will explore the properties of data with a uniform distribution.

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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.

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} } } {}

Exercise 9

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.
    Figure 3
    Figure 3 (graph.png)
  • b. Find the probability. P ( X < 4 ∣ X < 7 . 5 ) P ( X < 4 ∣ X < 7 . 5 ) size 12{P \( X<4 \lline X<7 "." 5 \) } {} =

Solution

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

Exercise 10: Discussion Question

What has changed in the previous two problems that made the solutions different?

Quartiles

Exercise 11

Find the average age of the cars in the lot.

Solution

μ μ size 12{μ} {} = 5

Exercise 12

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 4
    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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