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Continuous Random Variables: Practice 1

Module by: Dr. Barbara Illowsky, Susan Dean

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

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 1

The age of cars in the staff parking lot

Exercise 2

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

Solution 2

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 3

Continuous

Exercise 4

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

Solution 4

0.5 - 9.5

Exercise 5

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

Solution 5

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 6

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 7

  • 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 8

  • 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 9

  • b - 3 . 5 7 3 . 5 7 size 12{ { {3 "." 5} over {7} } } {}
Discussion Question

Exercise 10

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 11

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 12

  • b. k k size 12{k} {} = 7.25

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