Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 9 / a / ii / Solution

For a probability density function $f$, the finite expected value isThe absolute integrability condition makes the signed integral unambiguous. One can also allow an extended expectation when exactly one of the positive and negative parts has infinite integral, obtaining $+\infty$ or $-\infty$. If both parts have infinite integral, the expectation is undefined; subtracting two infinities is not a definition.