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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 $f(x)=\frac{1}{\mu}{e}^{-x/\mu}$ . 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 $(X=k)=\frac{{\lambda}^{k}{e}^{-k}}{k!}$ . 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 ).
Exponential: X ~ Exp ( m ) where m = the decay parameter
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 standard deviation?
State the probability density function.
f ( x ) = 0.2e ^{-0.2 x }
Graph the distribution.
Find P ( x >6).
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?
Find P ( x <4).
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?
In words, define the random variable X .
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).
Find the percentage of carbon-14 lasting longer than 10,000 years.
Thirty percent (30%) of carbon-14 will decay within how many years?
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