Why does $\mathbb{E}[X] = \sum^\infty_{r=0}\mathbb{P}(X > r)$?
13113
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
129584
How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$
73450
Show that $\sum_{n=1}^{\infty}X_n<\infty$ almost surely if and only if $\sum_{n=1}^{\infty}\mathbb E[\frac{X_n}{1+X_n}]<\infty$.
47211
Does $\varphi(x)\le\int_0^x\varphi(t)dt$ for all $x\in[0,\infty)$ imply $\varphi\equiv0$.
92810
What is $\sum_{r=1}^\infty\frac{r+2}{2^{r+1}(r)(r+1)}$?
34496
Proof that $e=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}$
141685
Why is expectation defined by $\int xf(x)dx$?
107842
Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$
118091
Taylor expansion of $\exp(-x)$: proof tail is positive
61963
Proving that $\sum\limits_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$
79024
the derivation of E(X|X)
27788
Why is the expected value $E(X^2) \neq E(X)^2$?
23837
Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?
19072
$\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
2722
Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.
36417
Why does $\mathbb{E}(\frac{X_1+...+X_n}{n})=\mathbb{E}(X)$?