Question 2 Suppose that we have data y = (v₁.....yn). Each data-point y, is assumed to be generated by a distribution with the following probability density function: 0² p(y₁|0) = - = exp(-)₁ 3₁ ² , y ≥ 0. The unknown parameter is 0, with > 0. (a) Write down the likelihood for 8 given y. Find an expression for the maximum likelihood estimate (MLE) 8. (b) A Gamma(a, 3) distribution is chosen as the prior distribution for 8. Derive the resulting posterior distribution for 8 given y. (c) We would like to choose the gamma prior distribution parameters so that a = 1, and P(0> 50+B) 0.05, B=2 Find the value of 3 that is needed. (d) The data are y =(4,4,8,8,4, C+3), C in the last Jigit olyan Inshas with n = 6. C = 3 (i) What is the MLE ? (ii) Using the prior distribution from part (c), what are the parameters of the posterior distribution for ? (iii) What are the posterior mean and standard deviation for A?

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