Confidence interval for Gamma parameter.

Question

Suppose $X \sim G(1, \theta^{2})$ and $Y \sim G(2, \theta^{2})$. How would I go about finding the constant $k$ for which $$P_{\theta}\left(\theta \leq k \sqrt{X+Y} \right) = 0.95$$

I figure that $X+Y \sim G(3, \theta^{2})$, but I'm not sure how to proceed with this observation. I'd appreciate any help.

Answer 1

Hint: $\theta\leq k\sqrt{X+Y}\iff X+Y\geq(\theta^2/k^2)$. You know the distribution of $X+Y$ so the answer should follow shortly.