Exponential

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Exponential

• The exponential distn is defined for xgt0
• Pdf ?e-?x
• CDF 1-e-?x
• Note CDF(0)0 and lim CDF 1
• The parameter ? is the parameter from the Poisson
  distn
• Units events/unit time

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Exponential

• Mean 1/?
• SD mean
• Note that we can think of units of mean and SD as
  being time, not events

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Exponential

• Memoryless property
• If time to next event has exp distn, then
  Prob(timegtAB timegtB) Prob(timegtA)
• If we wait for time B, then the prob of waiting
  an additional time A is the same as the original
  prob of waiting A

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Exponential

• Cond prob e-9/r / e-7/r
• e-2/r
• Prob(waitgt2)
• If we have waited 7 mins for elevator, then prob
  of waiting 2 more mins is same as when we walked
  up
• If we were willing to wait 2 mins then, we should
  be willing to wait 2 mins now
• Once we decide to wait for elevator, we never
  give up

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Exponential

• The exponential distn is the waiting time for the
  next event in a Poisson process
• If we have the waiting time for N separate
  events, we can think of that as the waiting time
  for the N-th event in a Poisson process
• Even if there is a gap between the times, this
  does not cause a problem due to the memoryless
  property

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Exponential

• Laplace distn 4.2.5, p. 213
• Sec 4.3 on Gamma distn
• 2 parameter family
• Contains the exponential and Erlang as special
  cases
• Important case Chi-square distn
• Used in statistics

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Exponential

• Gamma fn complete Gamma
• G(x) ? xk-1 e x dx
• Generalization of factorial
• G(x) (x-1) G(x-1)
• G(x) (x-1)! If x is an integer
• G(1/2) ?p

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Exponential

• Gamma is useful for modeling fns over positive
  domain