Show that $ E\Big[\sum_{i=1}^{n}(X_i - \bar{X_n})^2\Big] = (n-1)\sigma^2 $
41930
Mean of $ \sum (X_i - \bar{X})^2$
147705
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
5650
Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$
67489
How to show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$?
131626
Finding the distribution of $\|X-\mu \|_\Sigma^2$ with $X \sim N(\mu,\Sigma)$
48642
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) $
134696
How can I show that the conditional expectation $E(X\mid X)=X$?
140614
Show that $\sum^\infty_{n=m}\frac{1}{n^2} = O(\frac{1}{m})$
37283
Show that $\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment functions
73450
Show that $\sum_{n=1}^{\infty}X_n<\infty$ almost surely if and only if $\sum_{n=1}^{\infty}\mathbb E[\frac{X_n}{1+X_n}]<\infty$.
151996
Calculate the expected value of $Y=e^X$ where $X \sim N(\mu, \sigma^2)$
149362
Prove that $ \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy,$
127047
How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
57203
Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.
61963
Proving that $\sum\limits_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$
22759
The pdf of $\frac{X_1-\bar{X}}{S}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$