11_Continuous Probability Distributions Part 2-1

Px2 e ? x e 32 e 6 00025 px5 e ? x e 35 e 15 2231

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P(X>2) = e - λ x = e-3*2 = e -6 = 0.0025 P(X>.5) = e - λ x = e-3*.5 = e -1.5 = .2231 Exponential distribution example: Sci-fi insurance II
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31 4. What is the average time between accidents in weeks ? (Assume 1 month = 4 working weeks) 5. What is the mean number of accidents per week? 6. What is the probability of no accident next week? 3 accidents/mo = 3 accidents/ 4 weeks So, time between accidents is 4 weeks/ 3 accidents = 4/3 weeks 3 accidents/mo = 3 accidents/ 4 weeks = 0.75 accidents/week Use λ =.75 P(X>1) = e - λ x = e-.75*1 = e -.75 = .4724 Exponential distribution example Sci-fi insurance II
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32 Exponential distribution in Excel: X~ Expon()
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33 Exponential distribution: Example: Domino’s delivery guarantee X~Expon(λ = 3) (deliveries/hr) 1. Find the average and standard deviation of delivery time. 2. Domino’s is interested in reviving the famous money back guarantee for a late delivery. How long should the length of the grace period be if Domino wants to give money back to no more than 10% of delivery orders? 1/(3 deliveries/hr) = 1/3 hr or 20 minutes P(X>x) = 0.10 = e- λ x We know that λ =3, what should x be? ln(0.10)= - λ x = -3x => x=.7675 hr = 46 minutes
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34 Time 0 I 1 I 2 I 2 I 4 t λ : average “arrival rate” 1/λ: average “inter-arrival time” If the time between arrivals is exponentially distributed with rate λ, then the number of arrivals up to time t ~ Poisson(λ,t) Exponential distribution: Properties and implication How many people by t? N ~ Poisson When will customer j arrive? (from the last arrival) I j ~ Expon( λ ) Most queueing models use this! ! ) ( ) ( x e t x N P t x λ λ - = = s j e s I P λ - = ) (
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35 Example: National Car Rental at 30th St. Station λ = 10 customers per hour N = # of customers arriving in the first hour (Poisson vs Expon) I j : Time between j -1st and j th customer. The probability that no one arrives in one hour. N~Poisson( ) I j ~ Exponential ( ) P(N =0) P( I1 > 1) = λ , t = e- λ t( λ t)0/0! = e- λ λ e- λ x = e - λ Could restate as “probability that next customer arrives more than one hour from now.”
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36 Exponential distribution: Memoryless property Memoryles s : If X~ Expon(), how long the event (or the arrival) has not occurred is useless info! Do not have to remember the past to describe future. Examples Time between goals in a soccer match during the World Cup Time between arrivals at a bank teller Time between arrivals at Starbucks Exponential is the only continuous distribution with memorylessness. ) ( ) ( ) ( ) ( ) , ( ) | ( ) ( s X P e e x X P s x X P x X P x X s x X P x X s x X P x s x = = + = + = + - + - λ λ
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37 Exponential distribution: Memoryless property example: Banana J oe Example: The average time between arrivals of a Banana J oe’s party bus is 0.5 hour (30 minutes) 2. Now suppose the inter-arrival time is exponentially distributed.
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