Does such a statistical distribution exist? (Except exponential distribution.)

Question

Assume that continuous random variables $x_1,x_2,\cdots, x_N$ follows the same distribution with $x$ and they are statistically independent of each other, then $$ y=a_1x_1+a_2x_2+\cdots+a_N x_N, $$ where $a_1,a_2,\cdots>0$ and $N$ is an integer. If the survival function of $y$ is required to have a closed form, does such a distribution of $x$ exist?If any, how many distributions of $x$? If not, what is the distribution of $x$ in the special case $a_1=a_2=\cdots=1$?

Answer 1

If the distribution is discrete, the CDF has a closed form, as will a continuous r.v. have, especially if the $x_i$ are independent. The closed form may not be simple, but can be written even very generally.