Show that $E(X)=\sum_{n\in\mathbb{N}}P(X\ge n)$
14764
Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$
122115
Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
64720
Expected value of sums
10968
Let $X \sim Bin(n,p)$ show that $X/n \nsim Bin()$
34907
Show that $\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$
122678
Find the probability generating function of $Z=\sum_{i=1}^{N}X_i.$
90055
Show that $\mathcal P(x)\notin x$
91081
Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$
70129
Prove that in an ordered field $(1+x)^n \ge 1 + nx + \frac{n(n-1)}{2}x^2$ for $x \ge 0$
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
78754
Showing that $E[X|X<x]$ is smaller or equal than $E[X]$ for all x
135219
On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
156901
Showing that $X\sim E[X\mid\mathcal{G}]$ implies $X=E[X\mid\mathcal{G}]$ almost surely
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
86635
Optimizing $(X_{n}+c)^{2}$ to show that $P(\overline{X}_{n}\geq \lambda) \leq E(X_{n}^{2})/(EX_{n}^{2}+ \lambda^{2})$
82621
Show $P\{\xi_1+\dots+\xi_n=1\}=(\sum_{i=1}^n \lambda_i)\Delta + O(\Delta^2)$ (Shiryaev's Probability page 44)
53263
Show $\sum\limits_{d|n}\phi(d) = n$.