I have that for $1 \leq i \leq n$, the mutually independent random variables
$$X_i \sim \text{Poisson}(\mu_i)$$
Then what is the distribution of $$Y \sim \sum_{i=1}^{n}i X_i$$ It looks a bit like an expectation, so I am interested to know if anything is known about it? Otherwise, the best we can do to obtain $P(Y = k)$ is to sum over integer partitions of $k$ with part numbers which are Poissonly distributed with means $\mu_1, \mu_2,\dots$ etc?