discrete random variable

Formally, a random variable \[X\] is discrete if and only if \[\sum_{x\in S_{0}}P(X=x)=1\] for a countable set \[S_{0}\] of points where \[P(X=x)>0\] for \[x\in S_{0}\] and \[P(X=x)=0\] for \[x\notin S_{0}\]. The set \[S_{0}=\left\{ x\in \mathbb{R}\mid P(X=x)>0 \right\}\] is a subset of the image of \[X\], denoted by \[X(\Omega)\].