Show that any discrete or continuous random variable $X$ satisfies $E(X) = \int_{0}^\infty P(X > x)dx - \int_{0}^\infty P(X < -x)dx$
110768
Expectation of a mixed random variable given only the CDF
149362
Prove that $ \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy,$
116396
Formula for expected value of continuous random variable without using density
50163
Continuity of the  Ito integral
73367
On $E[E[Y|X]|X]= E[Y|X]$
104229
Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}[\frac{1}{X};X>y]=0$.
108560
Proof of a Gaussian Integral property
13732
$||f||_p\to ||f||_\infty$ under general assumptions
4287
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
76607
Show that exists a continuous function in $E$ that is not bounded
98710
Finding $\int_0^{\infty}xe^{-\lambda x} \, dx$
127486
Probability of continuous random variable
25618
$x,y,z$ independent and uniform random on $[0,1]$, $P(x \geq  yz)$?
44270
$E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$
129584
How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$
813
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
68484
Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?