16. 322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde The poisson distribution Outline 1 Discussion of the significance of the distribution 2) Summarize the derivation of the distribution 3)The Poisson approximation to the Binomial distribution 1. Discussion of the significance of the distribution The Posson distribution is a discrete integer-valued distribution which is of great importance in practical problems. It is employed in the study of telephone traffi through a switchboard, of the emission of B rays from radioactive material, of the emission of electrons from heated filaments and photo-sensitive surfaces, of line surges in a power transmission system due to the throwing of switches, of demands for service in a store or stockroom, and others. Using the statistics of the Poisson distribution, one answers questions related to the number of telephone operators, or stock clerks, or turnstiles in the subway which are required to yield a certain quality of service -where the quality of service is defined in terms of the probability that a customer will have to wait, or the expectation of the time he must wait, or the like. In control system analysis, the Poisson distribution may well lead to an excellent statistical description of the inputs seen by a system, or the occurrence of disturbances, etc. This distribution applies-in rough terms-to situations in which events occur at random over an interval I and we wish to know the probability P(k, 1, n)of the occurrence of exactly k events in the subinterval Il, of I where a is the average rate of occurrence of the events- which may possibly vary over the interval Very often the interval is time: thus we ask the probability that exactly k electrons be emitted in a given length of time However, other dimensions may equally well apply -such as intervals of distance. They may even be multidimensional-units of surface area or volume An example of an interval of volume is given by the statistics of the number of bacteria in volume samples of blood. In the"aerodynamics" of very high altitudes we may very well employ the Poisson distribution to give the probability of occurrence of k atoms in a specified volume The conditions under which the Poisson distribution may be expected to apply are given by the assumptions under which the distribution is derived Page 8 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde The Poisson Distribution Outline: 1) Discussion of the significance of the distribution 2) Summarize the derivation of the distribution 3) The Poisson approximation to the Binomial distribution 1. Discussion of the significance of the distribution The Posson distribution is a discrete integer-valued distribution which is of great importance in practical problems. It is employed in the study of telephone traffic through a switchboard, of the emission of B rays from radioactive material, of the emission of electrons from heated filaments and photo-sensitive surfaces, of line surges in a power transmission system due to the throwing of switches, of demands for service in a store or stockroom, and others. Using the statistics of the Poisson distribution, one answers questions related to the number of telephone operators, or stock clerks, or turnstiles in the subway which are required to yield a certain quality of service – where the quality of service is defined in terms of the probability that a customer will have to wait, or the expectation of the time he must wait, or the like. In control system analysis, the Poisson distribution may well lead to an excellent statistical description of the inputs seen by a system, or the occurrence of disturbances, etc. This distribution applies – in rough terms – to situations in which events occur at random over an interval I and we wish to know the probability P k I (,, λ) of the occurrence of exactly k events in the subinterval I1, of I where λ is the average rate of occurrence of the events – which may possibly vary over the interval. Very often the interval is time: thus we ask the probability that exactly k electrons be emitted in a given length of time. However, other dimensions may equally well apply – such as intervals of distance. They may even be multidimensional – units of surface area or volume. An example of an interval of volume is given by the statistics of the number of bacteria in volume samples of blood. In the “aerodynamics” of very high altitudes we may very well employ the Poisson distribution to give the probability of occurrence of k atoms in a specified volume. The conditions under which the Poisson distribution may be expected to apply are given by the assumptions under which the distribution is derived. Page 8 of 10