query: When is $\mathbb{E}\left[\frac{1}{\sum X_{i}}\right] = \frac{1}{\mathbb{E}\left[\sum X_{i}\right]}$?
positive_passages.docid: 130854
positive_passages.text: Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$
negative_passages.docid: 150321
negative_passages.text: How do you prove Cov $\left( \bar{X} , X_i - \bar{X} \right) = 0$?
negative_passages.docid: 25654
negative_passages.text: E[X| X>Y] for independent X, Y ~ N(0,1)
negative_passages.docid: 122603
negative_passages.text: How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
negative_passages.docid: 8913
negative_passages.text: Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
negative_passages.docid: 147705
negative_passages.text: Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
negative_passages.docid: 64844
negative_passages.text: Finding $E(|X_1|\ |\max |X_i| )$
negative_passages.docid: 43538
negative_passages.text: Show $\lim_{n \to \infty} n^{-1} E \left( \frac{1}{X}1_{[X>n^{-1}]} \right) =0$
negative_passages.docid: 48642
negative_passages.text: Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $
negative_passages.docid: 26226
negative_passages.text: Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.
negative_passages.docid: 72506
negative_passages.text: An example s.t. $E(E(X|\mathcal F_1)|\mathcal F_2)\neq E(E(X|\mathcal F_2)|\mathcal F_1)$
negative_passages.docid: 155023
negative_passages.text: Is $\frac{1}x\cos\left(\frac{1}x\right)$ Lebesgue integrable on $[0,1]$?
negative_passages.docid: 64856
negative_passages.text: Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$
negative_passages.docid: 93994
negative_passages.text: Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $X_i$'s are i.i.d Pareto variables
negative_passages.docid: 75902
negative_passages.text: Expectation of $\frac{1}{x+1}$ of Poisson distribution
negative_passages.docid: 61659
negative_passages.text: How to show that $ \sum_{n = 0}^{\infty} \frac {1}{n!} = e$?
negative_passages.docid: 82961
negative_passages.text: Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.