1. The Exponential Distribution

2. 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

3. EXPONENTIAL DENSITY FUNCTION f(x) =For x  0 and μ >0, f(x) = (1/μ)e -x/μ f(x) =0For x < 0 f(x) = 0 μ –the mean of the distribution = μ (i.e. μ = the average time between arrivals) μ 2 –the variance, σ 2 = μ 2 μ –the standard deviation, σ = μ P(X < x) = 1 - e -x/μP(X < x) = 1 - e -x/μ –Can be calculated using: A scientific calculator Excel

4. Example Time between arrivals is exponentially distributed –Average time between arrivals is 15 minutes (i.e. μ =.25 hrs.)

6. Calculating Probabilities What is the probability the next arrival will occur: –More than 30 minutes (.5 hours) from now? P(X >.5) = e -(.5/.25) = e -2 =.135335P(X >.5) = e -(.5/.25) = e -2 =.135335 –Within the next 15 minutes (.25 hours)? P(X <.25) = 1 - e -.25/.25 = 1 - e -1 = 1-.367879 =.632121P(X <.25) = 1 - e -.25/.25 = 1 - e -1 = 1-.367879 =.632121 –Between 3 and 6 minutes from now (.05 and.10 hours)? P(.05<X<.10) = (1-e -.10/.25 )- (1-e -.05/.25 ) = (1-e -.4 )- (1-e -.2 ) = e -.2 – e -.4 =.818731 -.670320 =.148411P(.05<X<.10) = (1-e -.10/.25 )- (1-e -.05/.25 ) = (1-e -.4 )- (1-e -.2 ) = e -.2 – e -.4 =.818731 -.670320 =.148411

7. EXCEL Exponential Distribution =EXP(x) gives the value of e x =EXPONDIST(x, 1/μ,TRUE) gives 1- e -x/μ, the cumulative probability up to time x, P(X < x) –x must be expressed in the same time units as μ –1/μ is the reciprocal of the average time between events

8. =1-EXPONDIST(.5,C2,TURE)=EXPONDIST(.25,C2,TURE) =EXPONDIST(.10,C2,TURE)- EXPONDIST(.05,C2,TURE)

9. REVIEW Exponential Distribution Relation to Poisson Distribution Calculating Probabilities Using –Calculator –Excel