1. Problem: 1) Show that is a set of sufficient statistics 2) Being location and scale parameters, take as (improper) prior and show that inferences on ……

2. Let and be independent r.q. Problem: Comparison of means and variances and the sufficient statistics 1) Show that for the priors (improper) Consider the samplings H1: same (but unknown) variances: Comparison of variances: Comparison of means: H2: unknown and different variances: (Behrens-Fisher problem)

3. 1)Show that a probability matching prior with the parameter of interest is given by 2) Show that the posterior for the correlation coefficient is: PROBLEM: Correlation Coefficient of the Bivariate Normal Model sample correlation is a sufficient statistic for

4. PROBLEM: Poisson Distribution Consider and Show that and, in consequence PROBLEM: Binomial Distribution Consider and Show that Hint: Analize the behaviour of expanding around and considering the asymptotic behaviour of the Polygamma Function, the moments of the Distribution,…

5. PROBLEM: Negative Binomial a: number of failures until experiment is stopped (fixed) X: number of successes observed θ: probability of failure 1) Find the transformations and such that the new parameters are location and scale parameters and transform them back to get the corresponding (improper) prior 2) Obtain the Fisher’s matrix and the Jeffrey’s prior 3) Find the reference prior 4) Show that it is a Probability Matching Prior PROBLEM: Weibull Distribution

6. Data: Problem: Linear Regression Linear Model: Assume precisions are known and show that: Model: Take and obtain (with uncertainty in x and y)

7. 4) If and Problem: 1) Generate a sample : 2) Get for each case the sampling distribution of 3) Discuss the sampling distribution of in connection with the Law of Large Numbers and the Central Limit Theorem How is distributed? 5) If How is distributed? 6) If (assumed to be independent random quantities) How is distributed?

8. PROBLEMS: 1) Show that if then 2) Show that if (  generate exponentials) 1) Show that if then Gamma Distribution: Beta Distribution:

9. (3,1,0) (3,2,±1) Problem 3D: Sampling of Hidrogen atom wave functions Evaluate the energy using Virial Theorem (3,2,0)