Solution
= Solution
For an <absolutely continuous probability distribution>, its <expectation> is
$$
\boxed{\mathbb E(X)=\int_{-\infty}^{\infty}x f_X(x)\,dx}.
$$
This is finite when $\int_{-\infty}^{\infty}|x|f_X(x)\,dx<\infty$. More generally it is an extended real value if only one of the positive and negative parts diverges; it is undefined if both diverge. For nonnegative $X$ it always exists in $[0,\infty]$.