How to prove that $\mathbf{E}\eta = \int_{0}^{\infty}\mathbf{P}(\{\eta > z\})dz$ for arbitrary non-negative random variable?
14764
Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$
140790
Prove that $E(X) = \int_{0}^{\infty} P(X>x)\,dx = \int_{0}^{\infty} (1-F_X(x))\,dx$.
85818
$E(|X-Y|)$ for $X$, $Y$ i.i.d and uniform on $[0,1]$
14930
Prove that the average of iid Gaussian random variables is Gaussian
83464
Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$
75645
Proof of $\frac{Y^{\lambda}-\lambda}{\sqrt{\lambda}}\to Z\sim N(0,1)$ in distribution as $\lambda\to\infty$?
35277
Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true?
141286
When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?
74054
Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$
100642
Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$
102286
Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$
107058
Prove that $E(X\ln X)\le E X E\ln X$
105145
Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
61133
Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
25652
Intuitive explanation for $\mathbb{E}X= \int_0^\infty 1-F(x) \, dx$
123806
How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?
38266
How to prove that exponential kernel is positive definite?