4. [Limit Theorems] Suppose there is a random sample of n calendar days and for each day we observethe number of purchase orders recieved by an online market, {x1, x2, , ..., xn}. Assume that the numberof orders follows the Poisson distribution with the daily intensity parameter A.(a) Suggest a consistent estimator Â for the unobserved parameter A.(b) Using the Central Limit Theorem, derive the asymptotic distribution of Â as n → ∞o.(c) Can you suggest a transformation g(Â) for this estimator such that the asymptotic distributiondoes not depend on the latent parameter \?

Here's a breakdown of the key concepts involved in estimating the rate of events in a Poisson…

Answered: 4. [Limit Theorems] Suppose there is a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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1. i Now consider the M/M/1 queue with arrival rate λ and service rate µ and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of pn, the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution
The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean λ = 7. (a) Compute the probability that more than 12 customers will arrive in a 2-hour period. (b) What is the mean number of arrivals during a 2-hour period.
2
9. Consider a bank office where customers arrive according to a Poisson process with an average arrival rate of customers per minute. The bank has only one teller servicing the arriving customers. The service time is exponentially distributed and the mean service rate is μ customers per minute. It turns out that the customers are impatient and are only willing to wait in line for an exponential distributed time with a mean of 1/μ minutes. Assume that there is no limitation on the number of customers that can be in the bank at the same time. a. Construct a rate diagram for the process and determine what type of queuing system this correspond to on the form A1/A2/A3. b. Determine the expected number of customers in the system when λ = 1 and μ = 2. c. Determine the average number of customers per time unit that leave the bank without being served by the teller when λ = 1 and µ = 2.
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
(i) and (ii) please, thank you
Which of the following best describes the Central Limit Theorem? (A) All of these choices are correct (B) The sample mean is close to 0.50. (C) The underlying population is normal. (D) If the distribution of a random variable is non-normal, the sampling distribution of the sample mean will be approximately normal for samples n ≥ 30.
Suppose that the claim size distribution of an insurance portfolio follows a Pareto distribution of the form α+1 α f(x) β = B\B+x (i) Derive a formula for the rth moment, ar, of this Pareto distribution in terms of its (r-1) th moment, αr-1. Show your steps clearly with reasons. (ii) From this expression find α3 and α4 using the known result for the mean μ of the above Pareto distribution. (You may assume that α>r, where a is one of the two parameters of the Pareto distribution).
Assume that n independent count variables {X,,X2,...,X,} are identically distributed as X, ~ Poi(0) where i= 1,2,...,n and to estimate E(X,)= 0, consider the sample mean estimator ô = -5x,. n (b) Use the expectation of the distribution of S = nô to compute the expectation E(@) and the bias of this estimator. State whether this estimator unbiased. (c) Use the variance of the distribution of S= nô to examine var(Ô) and the standard error of this estimator. Use the variance and the bias of ê to compute the Mean Squared Error (MSE) of this estimator. Comment on the MSE behaviour in the asymptotic limit as n→o. State whether this estimator is consistent or (d) not.
Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials"t proposes a Poisson distribution for X. Suppose that u 4. (Round your answers to three decimal places.)
Solve all plz

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