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Chapter review

If X has an exponential distribution with mean μ , then the decay parameter is m = 1 μ , 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 ) = 1 μ e x / μ . 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 ) = λ 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 = 1 m
  • standard deviation σ = µ
  • percentile k : k = l n ( 1 A r e a T o T h e L e f t O f k ) ( m )
  • Additionally
    • 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:   P ( X = k ) = λ k e k k ! 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?

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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.

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What is m ? What does it represent?

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What is the standard deviation?

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State the probability density function.

f ( x ) = 0.2e -0.2 x

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Find P (2< x <10).

0.5350

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Find the 70 th percentile.

6.02

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

What is the probability density function?

f ( x ) = 0.75 e -0.75 x

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What is the cumulative distribution function?

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Find the 30 th percentile.

0.4756

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Which is larger, the mean or the median?

The mean is larger. The mean is 1 m = 1 0.75 1.33 , which is greater than 0.9242.

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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?

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Are the data discrete or continuous?

continuous

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In words, define the random variable X .

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What is the decay rate ( m )?

m = 0.000121

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The distribution for X is ______.

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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.
    This is a blank graph template. The vertical and horizontal axes are unlabeled.
  2. Find the probability. P ( x <5,730) = __________
  1. Check student's solution
  2. P ( x <5,730) = 0.5001
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Find the percentage of carbon-14 lasting longer than 10,000 years.

  1. Sketch the graph, and shade the area of interest.
    Blank graph with horizontal and vertical axes.
  2. Find the probability. P ( x >10,000) = ________
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Thirty percent (30%) of carbon-14 will decay within how many years?

  1. Sketch the graph, and shade the area of interest.
    This is a blank graph template. The vertical and horizontal axes are unlabeled.
  2. Find the value k such that P ( x < k ) = 0.30.
  1. Check student's solution.
  2. k = 2947.73
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Questions & Answers

express the confidence interval 81.4% ~8.5% in interval form
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null rejected
Pratik
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A one sample, one-tail t-test is conducted and the test statistic value is calculated to be 2.56. The degrees of freedom for the test are 10. Which of the following conclusions for the test would be correct?
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what is null Hypothesis
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Andrea
A quality control specialist took a random sample of n = 10 pieces of gum and measured their thickness and found the mean 7.6 and variance 0.01. Do you think that the mean thickness of the spearmint gum it produces is 7.5? (4 Points)
Omer
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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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