Is it possible to find a sort of closed form distribution function of this random variable:
$$ Z= \sum_{k=1}^{N}X_kB_kC $$ where $N \perp (X_k)_{k\in \mathbb{N}} \perp (B_k)_{k\in \mathbb{N}} \perp C $, i.e everything is mutually independent of each other.
And $N \sim \mathrm{pois}(a)$, $(X_k)$ is an i.i.d sequence where $X_1 \sim \Gamma(d,t)$, $(B_k)$ is an i.i.d. sequence where $B_1 \sim \mathrm{Binom}(1,p)$ and $C\sim \mathcal{N} (\mu , \sigma^2)$.
Any help would be appreciated.