I have a given number, $v$, given by $$v = \sum_{i=1}^N\left\{v_i \cdot \left[m \cdot \left(1 - b_i \cdot \mathbb{1}_{\geq w}(v_i)\right) - b_i \cdot \mathbb{1}_{< w}(v_i) \right] - a \cdot \mathbb{1}_{\geq w}(v_i) \right\}$$ where all the $v_i$ are normally distributed, $v_i \sim \mathcal N(\mu,\sigma^2)$ with $\mu$ and $\sigma$ known, and $N$, $m$, $\mathbf{b}$, $w$, and $a$ are known constants (with $\forall i :b_i\in \{0,1\}$). What is the pdf $p(v|N,m,\mathbf{b},w,a, \mu, \sigma)$?
Now suppose $N$ is not known but rather Poisson distributed with rate $\lambda$ and $b_i$ has a known pdf $p(b_i|v_i)$, what's $p(v|m,w,a,\mu,\sigma,\lambda)$?
More generally, given a variable $$x = \sum_{i=1}^N f(x_i)$$ where $x_i \sim \mathcal N(\mu,\sigma^2)$, is there a standard way to derive $p(x|N,f(\cdot),\mu,\sigma)$?