Question: A sensor was recently placed on road to detect vehicles on a Highway in Virginia. In total 41 vehicles were detected by the sensor and the corresponding arrival time is stored in the csv...

Question: A sensor was recently placed on road to detect vehicles on a Highway in Virginia. In total 41 vehicles were detected by the sensor and the corresponding arrival time is stored in the csv file. You will be able find the inter-arrival intervals (or the time difference between two consecutive vehicles) based on the given data. Total time from first to last vehicle is 2,743 seconds. Assume the inter-arrival intervals ?? have exponential ?? ?? ?? 1 - ?? distribution with parameter ?? : ?? (?? |?? ) = ?? , where i=1, 2,…n. Assume the prior of ?? has an inverse ?? ?? ?? ?? ?? - -?? -1 ?? gamma distribution with parameters ?? = 1 and ?? = 1: ?? (?? ) = ?? ?? . ?? (?? ) (a) Please find the posterior distribution of ?? . (b) Generate 10,000 Monte Carlo samples based on the posterior distribution of ?? and compare the empirical distribution based on the samples with its prior distribution. (Draw both distributions on the same plot. Do NOT submit your generated samples.) (c) What is the posterior predictive distribution of the inter-arrival time for another new vehicle? (d) Based on (c), please show the empirical distribution of the inter-arrival time for the new vehicle based on 2,000 Monte Carlo samples and calculated its mean value. (Do NOT submit your generated samples.)