Prove that if $E(X^2+Y^2) \lt \infty$, $E(X|Y)=Y$, $E(Y|X)=X$, then $P(X=Y)=1$
662
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely
47895
Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$
140790
Prove that $E(X) = \int_{0}^{\infty} P(X>x)\,dx = \int_{0}^{\infty} (1-F_X(x))\,dx$.
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 $x, y \ge 0$
147705
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
138962
$X$ is a random variable, if $\Bbb E(X^2)=1$ and $\Bbb E(X)\geq a>0$, prove that $\Bbb P(X\geq\lambda a)\geq(a-\lambda a)^2$ for $0\leq\lambda\leq 1$.
85818
$E(|X-Y|)$ for $X$, $Y$ i.i.d and uniform on $[0,1]$
63124
Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
13561
Show: $E(|X|)\leqslant\liminf_{n\to\infty} E(|X_n|)$
4287
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
23946
Prove the definitions of $e$ to be equivalent
64856
Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$
65319
For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$
99024
Let $\,E=\{(x,y) \in \Bbb R^2: |x| \le 1,|y|\le 1\}.$
101754
Transformation of Random Variable $Y = X^2$
81363
How to compute $P(X>Y)$ if $X$ and $Y$ are independent and exponentially distributed
66902
Prove that $\int_{-\infty}^{\infty} \frac{e^{2x}-e^x}{x (1+e^{2x})(1+e^{x})}\mathrm{d}x=\log 2$