2. [8 marks] Consider the statistical model $$ Y_{i j}=\mu+\tau_{i}+R_{i j), \quad R_{ij} \stackrel{\text { i.i.d }}{\sim} G(0, \sigma) $$ where $i=1, \ldots, t, j=1, \ldots, r$, and $\sum_{i=1}^{t} \tau_{i}=0 $ Suppose $\theta=\sum_{i} a_{i} \tau_{i}$ is a contrast with the corresponding estimator $\widetilde {\theta)=\sum_{i} a_{i} \bar{Y}_{i+}$. Show that $$ \frac{\tilde{\theta) -\theta}{\widetilde{\sigma) \sqrt{\sum_{i} a_{i}^{2} /r}} $$ is a pivotal quantity and derive, with detailed steps, its distribution SP.PB. 109