**QUESTION**

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1. The number of claims in a month for a portfolio of insurance policies is thought to follow the Negative-binomial distribution with pdf \[ f(x)=\frac{\Gamma(20+x)}{\Gamma(x+1) \Gamma(20)} p^{20}(1-p)^{x}, \quad x=0,1,2, \ldots \] A sample of 4 months had claim numbers of $15,25,20$ and 21. a) There is no prior information for $p$. Using an uninformative prior to derive the posterior distribution for $p$. [12 marks] b) Show that the mode of the Beta distribution with pdf \[ f(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}, \quad 0<x<1 \] is \[ \frac{\alpha-1}{\alpha+\beta-2} \] [Hint: To obtain the marks for this question you will need to prove this is the mode - see file 'mode of a distribution' on Moodle for how this is done. You may assume without proof that the solution you find is actually a maximum, so there is no need to check the second differential]. [10 marks] c) Find the Bayesian estimates of $p$ under all-or-nothing loss and quadratic loss. [3 marks] [Total 25 marks]

Please answer question c, will downvote if answer is copied from other existing answers and question c ignored.