The simple Poisson process is characterized by a constant rate at whi.pdf

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The simple Poisson process is characterized by a constant rate at which events occur per unit time. A generalization of this is to suppose that the probability of exactly one event occurring in the interval [ t , t + t ] is ( t ) t + o ( t ) . It can then be shown that the number of events occurring during an interval [ t 1 , t 2 ] has a Poisson distribution with parameter = t 2 t 1 ( t ) d t The occurrence of events over time in this situation is called a nonhomogeneous Poisson process. The article "Inference Based on Retrospective Ascertainment," J. Amer. Stat. Assoc., 1989: 360-372, considers the intensity function ( t ) = exp ( + t ) as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusion. Suppose that a = 2 and b = .6 (close to values suggested in the paper), with time in years. 2.a What is the expected number of events in the interval [ 0 , 4 ] ? In [ 2 , 6 ] ? 2.b What is the probability that at most 15 events occur in the interval [ 0 , .9907 ] ?.

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Background: Suppose we want to model the spread of an infectious disease through a population. In the previous assignment, we did this with a discrete model. Here, we'll formulate things as a continuous model. We'll keep track of the numbers of individuals susceptible to the disease S ( t ) , infected by the disease I ( t ) , and recovered from the disease R ( t ) at time t (measured in days). Infected individuals have a chance to transmit the disease upon contact with a susceptible individual at a rate > 0 . Infected individuals also recover from the infection at a rate > 0 . We assume that all N individuals in the population produce offspring at a rate b > 0 and these newborns add to the pool of susceptible individuals. Finally, individuals in all disease states can die due to natural causes at a rate equal to the birth rate b . This description can be captured by the following system of equations: d t d S = b N S I b S d t d I = S I I b I d t d R = I b R Problems Problem 1 [3 points]: Given that S , I , and R are measured in number of people and t is measured in days, what must be the units of b , , and ? Justify your answer. Problem 2 [3 points]: Let N ( t ) = S ( t ) + I ( t ) + R ( t ) be the total population size at any point in time t . If the total population size remains constant for all time (i.e., N ( t ) = N for all t , where N is a constant) we say we have a "closed population". Show that the population in our model is closed. (Hint: Think about the rate of change d t d N of the population size and what must be true for the population to be constant.).

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Show that Greenwoods formula reduces to the binomial variance formula. Solution Let S(t) be the probability that an item from a given population will have a lifetime exceeding t. For a sample from this population of size N let the observed times until death of N sample members be Corresponding to each ti is ni, the number \"at risk\" just prior to time ti, and di, the number of deaths at time ti. Note that the intervals between each time typically are not uniform. For example, a small data set might begin with 10 cases, have a death at Day 3, a loss (censored case) at Day 9, and another death at Day 11. Then we have (t1 = 3, t2 = 11), (n1 = 10, n2 = 8), and (d1 = 1, d2 = 1). The Kaplan–Meier estimator is the nonparametric maximum likelihood estimate of S(t). It is a product of the form When there is no censoring, ni is just the number of survivors just prior to time ti. With censoring, ni is the number of survivors less the number of losses (censored cases). It is only those surviving cases that are still being observed (have not yet been censored) that are \"at risk\" of an (observed) death.[3] There is an alternative definition that is sometimes used, namely The two definitions differ only at the observed event times. The latter definition is right-continuous whereas the former definition is left-continuous. Let T be the random variable that measures the time of failure and let F(t) be its cumulative distribution function. Note that Consequently, the right-continuous definition of may be preferred in order to make the estimate compatible with a right-continuous estimate of F(t)..

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