prove or disprove pivot quantity uniform distribution

Question

let $X_i\sim U(0,\theta)$ indepedent uniform variables for $1\leq i \leq 5$

prove or disprove:

$\frac{X_{1}+X_{2}}{\theta}$ is pivot quantity for $\theta$.

I start with $P(\frac{X_{1}+X_{2}}{\theta}\leq t)$ then $P(X_{1}+X_{2}\leq \theta t)$

but I am not sure how to solve it.

Answer 1

Notice that $\frac{X_i}{\theta} \sim U(0, 1)$, so distribution of $\frac{X_1}{\theta} + \frac{X_2}{\theta}$ is the sum of two independent random values with distributions $U(0, 1)$, thus distribution does not depend on $\theta$. So it is pivot.