Joint distribution of sum of random variables in bivariate distribution

Question

We have the following bivariate distribution: $$f(x,y) = e^{-(\theta x + y/\theta)}$$ for x,y >0. We would like to find the distribution of $$\left(\sum_{i=1}^n X_i, \sum_{i=1}^n Y_i\right)$$

I'm not sure how to even approach this problem. Any help would be appreciated.

Thanks

Answer 1

HINT

Sounds like $X$ and $Y$ are independent since the joint pdf factors into individual ones. Individually both are exponential with different parameters. What is the distribution of a sum of independent exponential random variables?