How to show $\mathbb{E}(X) = \int_0^{\infty} (1-F_X(x)) \, dx$ for a continuous random variable $X \geq 0$ without assuming $f_X$ exists?
38324
Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$
116396
Formula for expected value of continuous random variable without using density
13287
Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$
42193
Show that $E(Y^p)=\int_{0}^{\infty}px^{p-1}P(Y\geq x)dx$
73450
Show that $\sum_{n=1}^{\infty}X_n<\infty$ almost surely if and only if $\sum_{n=1}^{\infty}\mathbb E[\frac{X_n}{1+X_n}]<\infty$.
4287
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
104229
Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}[\frac{1}{X};X>y]=0$.
43538
Show $\lim_{n \to \infty} n^{-1} E \left( \frac{1}{X}1_{[X>n^{-1}]} \right) =0$
116044
Can one prove $\int f > 0$ for $f > 0$ without Lebesgue integration?
98710
Finding $\int_0^{\infty}xe^{-\lambda x} \, dx$
61133
Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
84413
Why is $\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt$ true?
64571
Prove that $\int_{E}f =\lim \int_{E}f_{n}$
79011
If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
48937
Intuition of pdf of a continuous random variable
105145
Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$