STA510: Statistical Modelling And Simulation - HPP - NHPP - Statistical Modelling And Simulation Asignment Writing Help

Internal Code: 3FAG

Statistical Modelling And Simulation Asignment Writing Help

TASK In problem 1 on mandatory assignment 1 we considered visitors arriving to a website and the amount of time they spent on the web site. We shall now look further into this problem. As on mandatory assignment 1 we still assume that the time (in minutes) visitors are active on the web site after logging on is gamma distributed with parameters ? = 2 and ? = 3. (Remember that in Rthe parameter ? is called the shape parameter and the parameter ? is called the scale parameter. ) Except when we look at shorter time intervals the assumption made in mandatory assignment 1 that the arrival of visitors follows a homogeneous Poisson process (HPP) is not realistic. Based on usage statistics a more reasonable model for the arrival of visitors is to assume that this process is a nonhomogeneous Poisson process (NHPP) with intensity where t is number of hours since midnight. (Notice that the time scale for the arrival intensity ishours while the time scale for the visitor time is minutes - you need to pay attention to this in point Explain how we can use the hit or miss method for Monte Carlo integration to approximatethe integral. Calculate an expression for the required number of simulations you have to do to be at least95% certain that your estimate is no more than 10 from the true answer. R Implement the hit or miss method from point a) in R. Simulate an estimate of the integral. Use the number of simulations required to have theprecision described in point a). Calculate the probability of having more than 1250 visitors during one day (24 hours). Dothis by using a built-in R function where the result of the integral estimation above is one of the inputs. Two ways of simulating data from NHPP models have been discussed in the lectures. Why isthe thinning method more suitable than the transformation method in this case? Suggest a reasonable choice for ?max in the thinning algorithm. With this choice, how large the proportion of the arrival times generated in the interval [0, 24] from the HPP with intensity ?max will be deleted in the thinning step (step 3 in the thinning algorithm described in the lecture notes)? The algorithm for simulating the number of customers in a queue system over time describedin the lecture notes can be used here to simulate the number of active visitors at the websiteduring the day. Explain how we can use output of this algorithm to: i) estimate the probability that the maximum number of active visitors exceeds a certain number a, and ii) estimate the median and the 5% and 95% quantiles of the number of active visitors at a certain time point t.