# 5.3 The exponential distribution  (Page 6/25)

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## Chapter review

If X has an exponential distribution with mean μ , then the decay parameter is m = $\frac{1}{\mu }$ , and we write X Exp ( m ) where x ≥ 0 and m >0 . The probability density function of X is f ( x ) = me -mx (or equivalently $f\left(x\right)=\frac{1}{\mu }{e}^{-x/\mu }$ . The cumulative distribution function of X is P ( X x ) = 1 – e mx .

The exponential distribution has the memoryless property , which says that future probabilities do not depend on any past information. Mathematically, it says that P ( X > x + k | X > x ) = P ( X > k ).

If T represents the waiting time between events, and if T Exp ( λ ), then the number of events X per unit time follows the Poisson distribution with mean λ . The probability density function of P X is $\left(X=k\right)=\frac{{\lambda }^{k}{e}^{-k}}{k!}$ . This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf( λ , k ). The cumulative distribution function P ( X k ) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf( λ , k ).

## Formula review

Exponential: X ~ Exp ( m ) where m = the decay parameter

• pdf: f ( x ) = me (– mx ) where x ≥ 0 and m >0
• cdf: P ( X x ) = 1 – e (– mx )
• mean µ = $\frac{1}{m}$
• standard deviation σ = µ
• percentile k : k = $\frac{ln\left(1-AreaToTheLeftOfk\right)}{\left(-m\right)}$
• P ( X > x ) = e (– mx )
• P ( a < X < b ) = e (– ma ) e (– mb )
• Memoryless Property: P ( X > x + k | X > x ) = P ( X > k )
• Poisson probability: with mean λ
• k ! = k *( k -1)*( k -2)*( k -3)…3*2*1

## References

Data from the United States Census Bureau.

Data from World Earthquakes, 2013. Available online at http://www.world-earthquakes.com/ (accessed June 11, 2013).

“No-hitter.” Baseball-Reference.com, 2013. Available online at http://www.baseball-reference.com/bullpen/No-hitter (accessed June 11, 2013).

Zhou, Rick. “Exponential Distribution lecture slides.” Available online at www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf‎ (accessed June 11, 2013).

Use the following information to answer the next ten exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp (0.2)

What type of distribution is this?

Are outcomes equally likely in this distribution? Why or why not?

No, outcomes are not equally likely. In this distribution, more people require a little bit of time, and fewer people require a lot of time, so it is more likely that someone will require less time.

What is m ? What does it represent?

What is the mean?

five

What is the standard deviation?

State the probability density function.

f ( x ) = 0.2e -0.2 x

Graph the distribution.

Find P (2< x <10).

0.5350

Find P ( x >6).

Find the 70 th percentile.

6.02

Use the following information to answer the next seven exercises. A distribution is given as X ~ Exp (0.75).

What is m ?

What is the probability density function?

f ( x ) = 0.75 e -0.75 x

What is the cumulative distribution function?

Draw the distribution.

Find P ( x <4).

Find the 30 th percentile.

0.4756

Find the median.

Which is larger, the mean or the median?

The mean is larger. The mean is $\frac{1}{m}=\frac{1}{0.75}\approx 1.33$ , which is greater than 0.9242.

Use the following information to answer the next 16 exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

What is being measured here?

Are the data discrete or continuous?

continuous

In words, define the random variable X .

What is the decay rate ( m )?

m = 0.000121

The distribution for X is ______.

Find the amount (percent of one gram) of carbon-14 lasting less than 5,730 years. This means, find P ( x <5,730).

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x <5,730) = __________
1. Check student's solution
2. P ( x <5,730) = 0.5001

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

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x >10,000) = ________

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

1. Sketch the graph, and shade the area of interest.
2. Find the value k such that P ( x < k ) = 0.30.
1. Check student's solution.
2. k = 2947.73

#### Questions & Answers

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