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Saturday, July 19, 2014

 

Poisson, Erlang and CLT

Suppose you call some hotline and you are the 56th person in line, excluding the person currently being served. Callers depart according to a Poisson process with a rate of 2 per minute. What is the probability you will have to wait for more than 30 minutes ?

There are at least 3 ways to approach this.

Poisson

P(take > 30 min):
  55k=0Pλ(k,τ)=55k=0(λτ)keλτk!=55k=0(230)ke230k!0.285491

Erlang

1P(take 30 min):
  1300λktk1eλt(k1)!dt=1300256t55e2t55!dt10.714509=0.285491
See here.

CLT

  μ=nλ=562=28σ2=nλ2=14P(T>30)=1P(30μσ)1ϕ(0.5345)0.2981

See Example 6.12 of Introduction to Probability, 2nd Edition.


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