Recall that the exponential distribution with parameter> 0 has density g(x) = de- , (> 0). We write X-Exp (2) when a random variable X has this distribution. The Gamma distribution with positive parameters a (shape), B (rate) has density h (1) < 24-7e A, (x > 0). and has expectation. We write X Gamma (a, B) when a random variable X has this distribution. Suppose we have independent and identically distributed random variables X1,..., Xn, that we model as coming from an exponential distribution with some unknown parameter > 0. In short, X1,..., X, Exp (X). i.i.d (a) Bayesian Estimation and Confidence Regions: Calculations 6 puntos posibles (calificables, resultados ocultos) Compute the Jeffreys prior J (4). If the answer is improper, enter your answer such that ay(1) = 1. For > 0, 1J) = 10 In a particular scenario, we have n = i...d 10 observations X1,..., X10 with X; = 12. Recall X1,..., Xn Exp(). Using the Bayesian approach with the Jeffreys prior nj (4) computed above as the prior, compute the posterior distribution (X|X1,..., X10). (Enter the posterior distribution in proportionality notation without worrying about the normalization factor.) (|X1,..., , X10) ) α Compute Bayes' estimator, which is defined in lecture as the mean of the posterior distribution. (Enter your answer accurate to at least 3 decimal places.) i Bayes = 0.8333 0.8333 Compute the maximum-a-posteriori (MAP) estimator. (Enter your answer accurate to at least 3 decimal places.) MAP Compute a one-sided Bayesian confidence region with level 0.05 that takes the form (a,0). That is, if a random variable X(X|X1,..., X10), we are finding a such that P(X> a) = 0.95. Similarly, compute a one-sided Bayesian confidence region with level 0.05 that takes the form (0,6). That is, if a random variable X(X|X1,..., X10), we are finding b such that P(0