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# Continuous Random Variables: Practice 1

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.

## 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?

Continuous

### Exercise 4

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

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.
• 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. 1 9 1 9 size 12{ { {1} over {9} } } {}
• b.iv. Age of Cars
• 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.
• 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.
• 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} } } {}

## 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

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.
• 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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