The Exponential Distribution

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• The Exponential Distribution

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EXPONENTIAL DISTRIBUTION

• If the number of events in time period t has a
  Poisson distribution, the time between events has
  an exponential distribution
• Thus the same assumptions for the Poisson
  distribution must hold for the exponential
  distribution
• The exponential distribution is a memoryless
  distribution

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EXPONENTIAL DENSITY FUNCTION

• For x ? 0 and µ gt0 , f(x) (1/µ)e-x/µ
• For x lt 0 f(x) 0
• the mean of the distribution µ
  (i.e. µ the average time between arrivals)
• the variance, s2 µ 2
• the standard deviation, s µ
• P(X lt x) 1 - e-x/µ
• Can be calculated using
• A scientific calculator
• Excel

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Example

• Time between arrivals is exponentially
  distributed
• Average time between arrivals is 15 minutes
  (i.e. µ .25 hrs.)

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Calculating Probabilities

• What is the probability the next arrival will
  occur
• More than 30 minutes (.5 hours) from now?
• P(X gt .5) e-(.5/.25) e-2 .135335
• Within the next 15 minutes (.25 hours)?
• P(X lt .25) 1 - e-.25/.25 1 - e -1
  1- .367879
  .632121
• Between 3 and 6 minutes from now (.05 and .10
  hours)?
• P(.05ltXlt.10) (1-e-.10/.25)- (1-e-.05/.25)
  (1-e-.4)- (1-e-.2) e-.2
  e-.4 .818731 - .670320
  .148411

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EXCELExponential Distribution

• EXP(x) gives the value of ex
• EXPONDIST(x, 1/µ,TRUE) gives 1- e-x/µ, the
  cumulative probability up to time x, P(X lt x)
• x must be expressed in the same time units as µ
• 1/µ is the reciprocal of the average time between
  events

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REVIEW

• Exponential Distribution
• Relation to Poisson Distribution
• Calculating Probabilities Using
• Calculator
• Excel