Integrable random variable ($\mathbb{E}|X|<\infty$)

A real or complex random variable $X$ is integrable when $\mathbb{E}|X|<\infty$. This is integrability with respect to its probability measure, and ensures that its expected value is finite. A bounded random variable is always an integrable random variable; so is the example taking $-1,2,8$, for which $\mathbb{E}|X|=2.7$.

For a real random variable, put $X_{+}=\max (X,0)$ and $X_{-}=\max (-X,0)$, its positive part and negative part. Each has a nonnegative Lebesgue integral, possibly infinite. The expression $\mathbb{E}[X]=\mathbb{E}[X_{+}]-\mathbb{E}[X_{-}]$ is defined when at least one of these two expected values is finite; if both are infinite, the expected value is undefined. Integrability makes both finite. A nonnegative random variable can have expected value $+\infty$, even though it is not an integrable random variable.

The standard Cauchy distribution is a useful warning. With its probability density function $f(x)=1/[\pi (1+x^2)]$, the positive contribution is ${\int }_0^{R}xf(x)dx=\log (1+R^2)/(2\pi )$, which tends to infinity; the negative contribution has the same magnitude. Integrating symmetrically over $[-R,R]$ gives zero for each $R$, but this cancellation does not define an expected value. The Cauchy sample mean has an unchanged distribution, showing why the integrability hypothesis in the strong law of large numbers matters.

For an integrable random variable, conditional expectation preserves integrability, and linearity of expectation can safely combine finite averages without forming $\infty -\infty$.

Let $Z$ have standard normal distribution, let $m\ge 1$ be an integer and let $\alpha \in \mathbb{R}$. A finite expected value of $Z{e}^{\alpha Z^{2m}}$ requires absolute integrability, which reduces to finiteness ofFor $m=1$ it is finite exactly when $\alpha <1/2$. For every $m\ge 2$ it is finite exactly when $\alpha \le 0$: the higher power dominates the negative quadratic when $\alpha >0$, and for $\alpha \le 0$ the integrand is bounded by $z{e}^{-z^2/2}$. In the finite cases the signed expected value is zero by symmetry. Otherwise each signed part has infinite integral; a formal symmetric cancellation is only a Cauchy principal value.