Status	Time
0	2.05
0	2.27
0	2.76
0	2.09
1	4.67
0	3.11
1	1.58
0	2.32
0	1.22
0	1.55
0	1.1
1	0.29
0	3.08
0	1.32
1	1
0	0.84
1	5.18
0	1.7
0	1.91
1	2.59
1	2.06
0	2.18
0	0.57
0	1.57
0	0.75
1	5.23
0	1.78
0	1.84
1	1.63
0	6.92
1	2.31
0	0.88
0	0.73
0	1.75
0	0.76
0	2.12
0	4.89
1	1.87
0	0.88
0	2.89
1	2.84
1	0.47
0	1.17
1	1.42
0	2.48
0	2.69
0	1.65
0	3.33
0	0.5
1	5.79

The most recent statistics from Fundsquire show that 60 percent of Australian start-up businesses fail within their first three years . Suppose survival data is collected from a random sample of 50 start-ups that started since 01 January 2015 and the operational status of each company is recorded as at 30 June 2022. The data file "StartUp.csv" contains two variables Status (X) - where X = 1 if company i is still operational as at 30 June 2022, and X = 0 if company i is no longer operational as at 30 June 2022. Time (Y) - the time (in years) to complete shutdown of the company if X; = 0, or the censoring time (c) if X = 1. Let Z, denote the true lifetime of company i. Assume Z; follows an Exponential distribution. The model is Y = Z Ci if Xi = 0 if X, = 1 (that is Z > C) iid Z1, ..., | = Exp(0) (1) So Y is the observed survival or censoring time and Z is the true (but not always directly observed) survival time. If the start-up fails before the study end date, then Y = Z. If the start-up is still operational at the study end date, then all we know is that Zi > Ci, and the observed life time is equal to the censoring time c. The parameter is the rate parameter for the start-up true survival time. In this problem, the rate parameter and some of the true survival times Z = (Z,..., Zn) are unknowns. Our goal is to estimate the posterior density p(0|x, y) (where y = (y, yn), x = (x1, ..., En)) (a) [3 marks] Derive Jeffrey's prior for for the Exponential sampling model Z, ..., Zn|0 id Exp(0). Is Jeffrey's prior a proper prior for this model? (b) [2 marks] Assuming the prior you obtained in part (a), derive the full conditional posterior distribution p(0|z, x, y). (c) [3 marks] Derive the full conditional posterior distribution p(Z;0, z-i, x, y). (d) [4 marks] Implement a Gibbs sampling scheme that approximates the joint posterior distribution of and Z given y and x using the conditional distributions you derived in parts (b) and (c). Insert your computer code here. (e) [3 marks] Provide autocorrelation and traceplots for and a selection of the Zi's belonging to censored units. Comment on these plots. Also report the effective sample sizes. (f) [2 marks] Provide a 95% posterior interval estimate for 0. What is the expected survival time of a recent start-up in Australia given the data? (g) [3 marks] Is the Exponential distribution a valid sampling distribution assumption for this data? Run some checks to support your answer. Suggest how the model assumptions could be modified if your checks are not satisfied.