query: Show that if X is a continuous r.v. and it takes only positive values then $E(X) =\int_{0}^{∞}Z P[X ≥ t] dt $
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: 24683
negative_passages.text: For i.i.d. $\{X_n\}$ proving $\frac1n\max\limits_{1\leqslant k\leqslant n}|X_k|\xrightarrow{\mathrm{a.s.}}0\Leftrightarrow E(|X_1|)<+∞$
negative_passages.docid: 104229
negative_passages.text: Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}[\frac{1}{X};X>y]=0$.
negative_passages.docid: 126227
negative_passages.text: continuity of a probability measure if $\mu (\{x\})=0$
negative_passages.docid: 134696
negative_passages.text: How can I show that the conditional expectation $E(X\mid X)=X$?
negative_passages.docid: 50163
negative_passages.text: Continuity of the  Ito integral
negative_passages.docid: 64856
negative_passages.text: Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$
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: 5350
negative_passages.text: Explain why $E(|X|^p)=p\int_{0}^{\infty}y^{p-1}P(|X|>y) dy$
negative_passages.docid: 57695
negative_passages.text: Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
negative_passages.docid: 44270
negative_passages.text: $E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$
negative_passages.docid: 73367
negative_passages.text: On $E[E[Y|X]|X]= E[Y|X]$
negative_passages.docid: 40930
negative_passages.text: Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$
negative_passages.docid: 16212
negative_passages.text: If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
negative_passages.docid: 40564
negative_passages.text: Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
negative_passages.docid: 27705
negative_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 83873
negative_passages.text: Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$