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Unformatted text preview: IE 300/GE 331 Lecture 10 Negar Kiyavash, UIUC 1 Previous lecture • Poisson process – # of events that occur randomly in a time (spatial) unit is modeled by a Poisson r.v. X – Average # of events in the time (spatial) unit is known as λ (arrival rate: λ arrivals/time unit) – Poisosn process: the # of events X t occur in t time units ... 2 , 1 , , ! ) ( ) ( = = = = − x e x x X P x f x λ λ ... 2 , 1 , , ! ) ( ) ( = = = − x e x t x X P t x t λ λ IE 300/GE 331 Lecture 10 Negar Kiyavash, UIUC 2 Poisson distribution (cont) • Example: The number of patients arriving at the emergency room of a local hospital is modeled as a Poisson r.v. Suppose patients arrive at the rate of 2 every 30 minutes. – What is the probability that more than 1 patients arrive in 30 minutes? – What is the probability that 4 patients arrive in an hour? – What is the probability that in 4 onehour periods, at least one of them has 4 patient arriving IE 300/GE 331 Lecture 10 Negar Kiyavash, UIUC 3 IE 300/GE 331 Lecture 10 Negar Kiyavash, UIUC 4 Poisson distribution (cont) • Let X be the number of patients arriving in 30 minutes – X has a Poisson distribution with parameter λ =2 – Interested in P(X>1) – P(X>1)=P(X=2)+P(X=3)+… – Consider its complement: P(X>1)=1P(X ≤ 1)=1P(X=0)P(X=1) – P(X>1)=10.1350.271=0.594 135 . 1 1 ! ) ( 2 = = = = − − e e X P λ λ 0.271 1 2 ! 1 ) 1 ( 2 1 = = = = − − e e X P λ λ IE 300/GE 331 Lecture 10 Negar Kiyavash, UIUC 5 Poisson distribution (cont) • Probability that 4 patients arrive in an hour – Unit is different! One hour = 2*(30 minutes) – Let Y be the number of patients arriving in one hour 195 ....
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 Spring '09
 NegarKayavash
 Probability theory, probability density function, Negar Kiyavash

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