 Solution

The expectation of a real random variable $X$ on a probability space is its integral against the probability measure:For a discrete random variable, this becomes $\mathbb{E}X={\sum }_{x}x\mathbb{P}(X=x)$; for a random variable with probability density function $f$, it becomes $\mathbb{E}X={\int }_{-\infty }^{\infty }xf(x)dx$. A finite expectation requires $\mathbb{E}|X|<\infty$, making $X$ an integrable random variable. An extended expectation can also be defined when just one of the positive and negative parts has finite expectation; the indeterminate difference $\infty -\infty$ is not allowed.