Exercise 1.2 (*** Shifted exponential (50%)). Let 0 ~ utoExp(1) be a shifted exponential distribution with location parameter u ER and scale parameter o ER+. Its PDF is f(0) e-(0-1)/01(0 > u). = -e We can easily generate n IID copies in R by using the syntax theta mu+sigmaxrexp (n). Assume that IID [21, ..., In, In+1 | 0] e 1 + Exp(1)/0; 0 Exp(1), where oo is a known constant. 1. (10%) Are 0(x1 - 1) and 0(Xn+1 1) independent? Explain. 2. (10%) Find the posterior distribution [0 | 21:n). 3. (10%) Find the prior predictive distribution (Xn+1]. 4. (10%) Find the posterior predictive distribution (In+1 | 21:n). 5. (10%) Suppose oo 9/10 and 21:9 = (1.028, 1.473, 1.103, 1.096, 1.077, 1.089, 1.058, 1.015, 1.065)T. Using R to plot the PDF s of the following distributions on the same graph. Interpret. (Use no more than 30 words.) [21], [2221], [23 | 21:2), [24 | 21:3), [210 21:9) Exercise 1.2 (*** Shifted exponential (50%)). Let 0 ~ utoExp(1) be a shifted exponential distribution with location parameter u ER and scale parameter o ER+. Its PDF is f(0) e-(0-1)/01(0 > u). = -e We can easily generate n IID copies in R by using the syntax theta mu+sigmaxrexp (n). Assume that IID [21, ..., In, In+1 | 0] e 1 + Exp(1)/0; 0 Exp(1), where oo is a known constant. 1. (10%) Are 0(x1 - 1) and 0(Xn+1 1) independent? Explain. 2. (10%) Find the posterior distribution [0 | 21:n). 3. (10%) Find the prior predictive distribution (Xn+1]. 4. (10%) Find the posterior predictive distribution (In+1 | 21:n). 5. (10%) Suppose oo 9/10 and 21:9 = (1.028, 1.473, 1.103, 1.096, 1.077, 1.089, 1.058, 1.015, 1.065)T. Using R to plot the PDF s of the following distributions on the same graph. Interpret. (Use no more than 30 words.) [21], [2221], [23 | 21:2), [24 | 21:3), [210 21:9)