If $x_1$ and $x_2$ are the values of random sample of size $n=2$ from a population having a uniform distribution with parameters $0$ and $\theta>0$, find the number $k$ so that the interval $[0, k(x_1+x_2)]$ is a $(1-\alpha)100\%$ confidence interval for the parameter $\theta$ when (1) $\alpha\leq 1/2$; and (2) $\alpha>0$.
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You need to determine the distribution of $X_1 + X_2$ in function of $\theta$. Then calculate $k_\alpha$ such that $$ \mathbb P ( k_\alpha (X_1+ X_2) \geq \theta) = 1-\alpha. $$