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Continuous Random Variables: Practice 2

Module by: Dr. Barbara Illowsky, Susan Dean

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

Student Learning Outcomes

  • The student will explore the properties of data with a 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?

Solution 2

Continuous

Exercise 3

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

Solution 3

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

Exercise 4

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

Solution 4

m m size 12{m} {} = 0.000121

Exercise 5

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

Solution 5

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.
    Figure 1
    Blank graph with vertical and horizontal axes.
  • b. Find the probability. P ( X < 5730 ) P ( X < 5730 ) size 12{P \( X<"5730" \) } {} =

Solution 6

  • 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.
    Figure 2
    Blank graph with horizontal and vertical axes.
  • b. Find the probability. P ( X > 10000 ) P ( X > 10000 ) size 12{P \( X<"5730" \) } {} =

Solution 7

  • 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.
    Figure 3
    Blank graph with vertical and horizontal axes.
  • 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 8

  • b. k k size 12{k} {} = 2947.73

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