query: Let $X$ be a positive random variable with distribution function $F$. Show that $E(X)=\int_0^\infty(1-F(x))dx$
positive_passages.docid: 14764
positive_passages.text: Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$
negative_passages.docid: 138962
negative_passages.text: $X$ is a random variable, if $\Bbb E(X^2)=1$ and $\Bbb E(X)\geq a>0$, prove that $\Bbb P(X\geq\lambda a)\geq(a-\lambda a)^2$ for $0\leq\lambda\leq 1$.
negative_passages.docid: 124463
negative_passages.text: Integral from $0$ to $\infty$ of $\ln(x)/e^x$
negative_passages.docid: 97039
negative_passages.text: Use integral test to show that Gaussian Integral $\int_{-\infty}^{\infty} e^{-x^2} \ dx$ converges?
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
negative_passages.docid: 27511
negative_passages.text: Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$
negative_passages.docid: 125723
negative_passages.text: Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$
negative_passages.docid: 107058
negative_passages.text: Prove that $E(X\ln X)\le E X E\ln X$
negative_passages.docid: 43538
negative_passages.text: Show $\lim_{n \to \infty} n^{-1} E \left( \frac{1}{X}1_{[X>n^{-1}]} \right) =0$
negative_passages.docid: 92569
negative_passages.text: Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable
negative_passages.docid: 35325
negative_passages.text: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
negative_passages.docid: 34278
negative_passages.text: A basic question on expectation of distribution composed random variables
negative_passages.docid: 141685
negative_passages.text: Why is expectation defined by $\int xf(x)dx$?
negative_passages.docid: 34496
negative_passages.text: Proof that $e=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}$
negative_passages.docid: 46844
negative_passages.text: Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$
negative_passages.docid: 102399
negative_passages.text: Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $
negative_passages.docid: 61659
negative_passages.text: How to show that $ \sum_{n = 0}^{\infty} \frac {1}{n!} = e$?