16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde Example: Telephone calls throughout a business day 9am 12pm 5pm where a is the average number of phone calls per minute P(exactly k events in an interval (ti, L))=P(k)=ue- where u=2(r)dr 2. Summarize the derivation of the distribution In an analytic derivation of the distribution, the following assumptions are made a. The occurrence of events in all non-overlapping intervals are independent. b. The probability of the occurrence of 1 event in an infinitesimal interval dI is an infinitesimal of the order of dl; the probability of occurrence of more than one event in that interval is an infinitesimal of higher order than dI These assumptions permit the derivation of the distribution; thus if the circumstances of a given situation seem to meet these conditions, the Poisson distribution may be expected to apply The derivation proceeds by determining the probabilities for 0 and 1 event which establish an apparent functional form for the distribution, then by proving the form by showing in general that the form holds for 0 and 1 and that if it holds for n, it holds also for n+I The derivation is developed in adequate detail in the text, and leads to the general form of the Poisson distribution P(h)=ue, where u=a(r)dr This is the probability of exactly k events in the interval (t, t2) in which i(t)is the of events over the interval Page 9 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Example: Telephone calls throughout a business day where λ is the average number of phone calls per minute. () = 1 µk e−µ P(exactly k events in an interval (t , t )) = P k 12 k ! t2 where µ = ( )d ∫ λ τ τ . t 1 2. Summarize the derivation of the distribution In an analytic derivation of the distribution, the following assumptions are made: a. The occurrence of events in all non-overlapping intervals are independent. b. The probability of the occurrence of 1 event in an infinitesimal interval dI is an infinitesimal of the order of dI; the probability of occurrence of more than one event in that interval is an infinitesimal of higher order than dI. These assumptions permit the derivation of the distribution; thus if the circumstances of a given situation seem to meet these conditions, the Poisson distribution may be expected to apply. The derivation proceeds by determining the probabilities for 0 and 1 event which establish an apparent functional form for the distribution, then by proving the form by showing in general that the form holds for 0 and 1 and that if it holds for n, it holds also for n+1. The derivation is developed in adequate detail in the text, and leads to the general form of the Poisson distribution: () = 1 µ e−µ , where µ = ∫ t2 Pk λ τ τ k ( )d k ! t 1 This is the probability of exactly k events in the interval (t1,t2) in which λ( )t is the average rate of occurrence of events over the interval. Page 9 of 10