# 5.6 Practice 2: exponential distribution

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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

What is being measured here?

Are the data discrete or continuous?

Continuous

In words, define the Random Variable $X$ .

$X$ = Time (years) to decay carbon-14

What is the decay rate ( $m$ )?

$m$ = 0.000121

The distribution for $X$ is:

$X$ ~ Exp(0.000121)

## Probability

Find the amount (percent of 1 gram) of carbon-14 lasting less than 5730 years. This means, find $P\left(x<\text{5730}\right)$ .

• Sketch the graph. Shade the area of interest.
• Find the probability. $P\left(x<\text{5730}\right)$ =
• $P\left(x<\text{5730}\right)$ = 0.5001

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

• Sketch the graph. Shade the area of interest.
• Find the probability. $P\left(x>\text{10000}\right)$ =
• $P\left(x>\text{10000}\right)$ = 0.2982

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

• Sketch the graph. Shade the area of interest.
• Find the value $k$ such that $P\left(x .
• $k$ = 2947.73

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