If $x\geq 0$ is a non-negative real number, then $$x=\int_0^\infty m(\{ x \geq t\})dt.$$ Let $X\geq0$ be a non-negative random variable on $\Omega$ and let $\mathbb P$ be probability measure on $\Omega$. Then an application of Fubini's theorem shows that the expectation of $X$, defined as $$\mathbb E[X]:=\int_\Omega X \ d\mathbb P,$$ is equal to $$\mathbb{E}[X] = \int_0^\infty \mathbb P ( \{ X > \lambda\} ) \ d\lambda,$$ (this bears a resemblance to the first display) which we may also write as $$\int_{\mathbb R^+} \lambda \mathbb P(\{ X>\lambda\}) \ \frac{d\lambda}\lambda,$$ viewed as the integral of the first moment of $\mathbb P(X > \lambda)$ with respect to Haar measure $\frac{d\lambda}\lambda$ on the Lie group $\mathbb R^+ = \{ x >0\}$.
My questions:
- Is there a meaningful interpretation of the quantity $$\lambda \mathbb P(X>\lambda)$$ in probability theory?
- Where is Haar measure $\frac{d\lambda}\lambda$ used in probability theory?
- Is there an interpretation of higher moments $$\lambda^p \mathbb P(X>\lambda)$$ of the probability distribution in probability theory?
Here is some context and some background:
- I know very little about Lie groups.
- The Haar measure is useful in some problems because it is invariant under scaling $x\mapsto cx$ and inversion $x\mapsto x^{-1}$; one can also view the Gamma function at some positive real $s>0$ as the $s$-th moment of the function $e^{-t}$ w.r.t. Haar measure, $$\Gamma(s) = \int_{\mathbb R^+}e^{-t}t^s \frac{dt}t.$$
More generally, if $Y$ is a measure space and $f$ is a non-negative measurable function on $Y$, then $$\| f\|_{L^1(Y)} = \int_{\mathbb R^+} \lambda m\{ f > \lambda \} \frac{d\lambda}\lambda,$$ and the same question can be made in this more general setting. Here, we are looking at the first moment of the distribution function of $f$ with respect to Haar measure. More generally, for $0<p<\infty$, we have $$p^{-1}\|f\|_{L^p(Y)}^p = \int_{\mathbb R^+}t^p\lambda_f(t)\frac{dt}t$$ where $\lambda_f(t) = m\{f>\lambda\}$ is the distribution function of $f$, so we see the $L^p$ norm is essentially the $p$-moment of the distribution function w.r.t. Haar measure.
A remark:
- There is also a physical interpretation of this question; this question could be interpreted as a connection between mass (the $L^1$ norm) and the first moment (see the second-to-last display).