Walser (1969) gave the following data on the month of giving birth for 700 women giving birth for the first time. The births took place at the University Hospital of Basel, Switzerland.  

<html><body><table><tr><td>Month</td><td></td><td>No. of births</td><td>Month</td><td>No. of births</td><td>Month</td><td>No. of births</td></tr><tr><td>1</td><td>January</td><td>66</td><td>5 May</td><td>64</td><td>9. September</td><td>54</td></tr><tr><td>2</td><td>February</td><td>63</td><td>6 June</td><td>74</td><td>10 October</td><td>51</td></tr><tr><td>3</td><td>March</td><td>64</td><td>7 July</td><td>70</td><td>11 November</td><td>45</td></tr><tr><td>4</td><td>April</td><td>48</td><td>8 August</td><td>59</td><td>12 December</td><td>42</td></tr></table></body></html>  

We have unknown parameters $\theta_{1},\dots,\theta_{12}$ where, given the values of these parameters, the probability that one of these births takes place in month $j$ is $\theta_{j}$ and January is month 1, February is month 2 and so on through to December which is month 12. Given the parameters, the birth dates are assumed to be independent.  

Our prior distribution for $\theta_{1},\dots,\theta_{12}$ is a Dirichlet distribution with parameters $a_{1}=a_{2}=$ $\dots-a_{12}=2$ .  

(a) Find the posterior distribution of $\theta_{1},\dots,\theta_{12}$ .  
(b) For each of $j=1,\dots,12$ , find the posterior mean of $\theta_{j}$ .  
(c) For each of $j=1,\cdot\cdot\cdot,12$ , find the posterior probability that $\theta_{j}>1/12$ and comment on the results.   
(d) Find the joint posterior distribution of $\theta_{1},\theta_{2},\Tilde{\theta}_{2}$ , where $\tilde{\theta}_{2}=1-\theta_{1}-\theta_{2}$ .  

Note: You may use R for the calculations but give the commands which you use with your solution.  