Theorem

Discrete Given a discrete RV $X$ with range $\mathscr{X}$ and pmf $p$, and given $g:\mathbb{R}\to\mathbb{R}$, then $$E[g(X)]=\sum_{x\in\mathscr{X}}g(x)\ p(x)$$Continuous Given a continuous RV $X$ with range $\mathscr{X}$ and pdf $f$, and given $g:\mathbb{R}\to\mathbb{R}$, then $$E[g(X)]=\int_{\mathscr{X}} g(x)f(x) \ dx$$ only if:$$E[|g(X)|]=\int|g(x)|f(x) \ dx<\infty$$