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Inside Collection (Textbook):

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

# Practice 2: Exponential Distribution

Summary: In this module the student will explore the properties of data with an exponential distribution.

## Student Learning Outcomes

• The student will analyze data following the exponential distribution.

## Given

Carbon-14 is a radioactive element with a half-life of about 5730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121 . We start with 1 gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

## Describe the Data

### Exercise 1

What is being measured here?

### Exercise 2

Are the data discrete or continuous?

Continuous

### Exercise 3

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

#### Solution

X X size 12{X} {} = Time (years) to decay carbon-14

### Exercise 4

What is the decay rate ( m m size 12{m} {} )?

#### Solution

m m size 12{m} {} = 0.000121

### Exercise 5

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

#### Solution

X X size 12{X} {} ~ Exp(0.000121)

## Probability

### Exercise 6

Find the amount (percent of 1 gram) of carbon-14 lasting less than 5730 years. This means, find P ( x < 5730 ) P ( x < 5730 ) size 12{P $$X<"5730"$$ } {} .

• a. Sketch the graph. Shade the area of interest.
• b. Find the probability. P ( x < 5730 ) P ( x < 5730 ) size 12{P $$X<"5730"$$ } {} =

#### Solution

• b. P ( x < 5730 ) P ( x < 5730 ) size 12{P $$X<"5730"$$ } {} = 0.5001

### Exercise 7

Find the percentage of carbon-14 lasting longer than 10,000 years.

• a. Sketch the graph. Shade the area of interest.
• b. Find the probability. P ( x > 10000 ) P ( x > 10000 ) size 12{P $$X<"5730"$$ } {} =

#### Solution

• b. P ( x > 10000 ) P ( x > 10000 ) size 12{P $$X<"5730"$$ } {} = 0.2982

### Exercise 8

Thirty percent (30%) of carbon-14 will decay within how many years?

• a. Sketch the graph. Shade the area of interest.
• b. Find the value k k size 12{k} {} such that P ( x < k ) = 0 . 30 P ( x < k ) = 0 . 30 size 12{P $$X<k$$ =0 "." "30"} {} .

#### Solution

• b. k k size 12{k} {} = 2947.73

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