A transformation of a discrete random vector

Question

I have a vector $X=(X_1,\ldots,X_n)$ of i.i.d. random variables with values in a discrete set $S \subset \mathbb{R}$ and a (continuous but not necessarily linear) function $F:\mathbb{R}^n\mapsto \mathbb{R}$. I would like to obtain the distribution of $F(X)$. The obvious way to do that is to write $$\mathbb{P}[F(X)=y]=\sum_{x_1,\ldots,x_n \in S}\mathbb{P}[X_1=x_1]\cdots\mathbb{P}[X_1=x_n]\mathsf{1}_{[F(x_1,\ldots,x_n)=y]}$$ for each $y \in \{z \in \mathbb{R}; z=F(x) \mbox{ for some } x \in S^n \}$. Is there a less clumsy way to calculate the distribution of $F(X)$? Thanks!