Let $X$ be a random variable with values in $\mathbb Z_{\geq0}$. Show that $E[X]=\sum_{k=0}^\infty P[X>k]$
92569
Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable
89544
Expectation of a random variable that takes points in the natural numbers
73367
On $E[E[Y|X]|X]= E[Y|X]$
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?
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
68097
Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$
105145
Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
21004
Formula similar to $EX=\sum\limits_{i=1}^{\infty}P\left(X\geq i\right)$ to compute $E(X^n)$?
114184
Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$
72748
Proof that $\sum\limits^{\infty}_{i=0}{iP(X > i)} = \frac12(E(X²)-E(X))$ for $X$ nonnegative and integer valued
61659
How to show that $ \sum_{n = 0}^{\infty} \frac {1}{n!} = e$?
83058
Finding $\mathrm{Var}(N)$ if $N=\inf\{n\ge1:\sum_{i=1}^nX_i>1\}$ where $X_i$'s are i.i.d Exponential variables
81302
Showing that $a_n \in \ell^{\infty}$
17558
How do I show that the sum of two random variables is random variable?
82279
Prove that it is a random variable iff it is constant on each partition
149792
Show that $\lim_{p \to 0} \mathbb {E} [(X^p-1)/p ] = \mathbb{E} [\log X]$
85818
$E(|X-Y|)$ for $X$, $Y$ i.i.d and uniform on $[0,1]$