4. (Conditional Probabilities, Finite Stochastic Processes; 25pts). Three machines A, B and C produce respectively 48%, 27% and 25% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 4% and 8%, respectively. (i). If an item is selected at random, find the probability that the item is defective. (ii). If an item is selected at random and is found to be defective, find the probability that the item was produced by machine A. (ii) Use R to simulate each of the processes in R and estimate the parameters and compare them with actual parameters in your models. Check the time series plot and acf plots for each process and examine each model carefully and report your observations. (i) (ii) (iii) (1 0.5B+.06B2)X = (1 B)Zt (1 -0.2B)X = Zt. (1 -0.036 B)X+= Zt 0.7Zt1. 3 Starting with the R function for the simulation of the immegration-death process, modify it to include "births in addition to immigrations and death. Use your modified function to simulate a bug realisation from the birth-immigration-death process where the birth, death, and immigration rates are all from an initial condition of zero. one, starting Exercises 1 Find the transition function of the two-state birth and death process by solving the forward equation. 0, 1, and 2, 2 Consider a birth and death process having three states H2. Use the forward equation and birth and death rates such that 20 = to find Poy(t), y = 0, 1, 2.