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2d
revised Cramer-Blackwell estimator for uniform distribution.
added 4 characters in body
2d
comment Cramer-Blackwell estimator for uniform distribution.
One of the $X_k$s is equal to $T_2$ and the $n-1$ others are uniform on $(0,T_2)$ with conditional mean $\frac12T_2$ hence $E(T_1\mid T_2)=\frac2n\left(T_2+(n-1)\frac12T_2\right)=\frac{n+1}nT_2$.
2d
comment Prove or Disprove Θ
"for large $n$" Actually, for every $n\geqslant1$.
2d
comment Convergence of sequences of random variable
@user1952009 Not sure this is even related.
2d
revised Norris exercise: Showing $P_0[\text{no return to}\ 0]=6/\pi^2$
edited tags; edited title
2d
comment Norris exercise: Showing $P_0[\text{no return to}\ 0]=6/\pi^2$
The usual approach is to compute $h_k^i=P(X\ \text{hits}\ i\ \text{before}\ 0\mid X_0=k)$ for every $0\leqslant k\leqslant i$. These involve a random walk on the finite graph from $0$ to $i$ hence one simply has to solve a linear system of size $(N-1)\times(N-1)$ with boundary conditions $h_0^i=0$ and $h_i^i=1$. The exact formula might be $$h_k^i=\frac{1^2+2^2+\cdots+k^2}{1^2+2^2+\cdots+i^2},$$ but you should check it, then one simply notes that $h_k^i\to h_k$ when $i\to\infty$, which simultaneously explains the results already in your post and answers your question.
2d
comment $\lim\limits_{K\to\infty} [n,K]$.
"My" definition? No, not mine, I am simply reproducing the canon... Limits for nondecreasing sequences of sets are defined in my first comment, for nonincreasing sequences $(A_n)$ of sets, consider $$\lim A_n=\bigcap_kA_k.$$
2d
comment $\lim\limits_{K\to\infty} [n,K]$.
No, once again limits of monotonous sequences of sets do not require a metric.
2d
comment The probability of the sum of $10$ dice rolls adding up to $57$
The solution probably rather states that the probability of the events are $ \binom{10}{3}\cdot\frac1{6^3}$, $ \binom{10}{1} \cdot \binom{9}{1}\cdot\frac1{6^3}$ and $\binom{10}{1} \cdot\frac1{6^3}$ respectively. Decidedly, your question needs a thorough revising.
2d
comment $\lim\limits_{K\to\infty} [n,K]$.
There is no underlying metrics involved, rather, one uses that, for every nondecreasing sequence of sets $(A_n)$, $$\lim A_n=\bigcup_kA_k.$$
2d
comment Find the radius of convergence
Then try it and you will see it gives readily the answer.
2d
comment Finding the limit of a sequence $\lim_{n \to \infty} 2^{2n+3}\left(\sqrt[3]{8^n+3}-\sqrt[3]{8^n-3}\right)$
Hint: Let $u(x)=\sqrt[3]{x}$, then, when $\epsilon\to0$, $$u(1+\epsilon)-u(1-\epsilon)\sim2u'(1)\epsilon.$$ You might want to compute $u'(1)$ and use all this for $$\epsilon=3\cdot8^{-n}.$$
2d
comment Intersection of dense sets in $\mathbb{N}$
@OlivierOloa How?
2d
comment Return time for two independent one dimensional random walks
Actually, $P(\tau_1>k)\sim1/\sqrt{\pi k}$ hence $E(\tau)$ is infinite.
2d
revised Infinite expectation implies infinite random variable?
edited tags
2d
comment Proof of the existence of a reversible stationary distribution
$$\sum_j\pi_jp(j,k)=c\sum_j\frac{p(i,j)p(j,k)}{p(j,i)}=c\sum_j\frac{p(i,k)p(k,j)‌​}{p(k,i)}=c\frac{p(i,k)}{p(k,i)}\sum_jp(k,j)=c\frac{p(i,k)}{p(k,i)}=\pi_k$$
2d
comment Isomorphism of a group $G = \left\langle x , y \mid x^5 = y^2 = e , x^2y = yx \right\rangle$
@DerekHolt "y2=e is not one of the relations of the presentation!" Yeah, and since x5=x2=e is absurd, and since the author of the question does not even correct it, you might want to direct your anger at them, not at this answerer.
2d
revised Use convolution theorem to evaluate $\int_0^\infty e^{-((|a+su|)/c)^b}e^{-(u/k)^p}du$
added 3 characters in body
2d
comment If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?
@Ron True. Note that the Great Man used other substances as well, but one cannot recommend these publicly I am afraid.
2d
comment If X and Z are independent and Y and Z are independent random variables, is cov(XY, Z) = 0?
The revised suggestion is wrong as well, try $(X,Y)$ uniform on $\{-1,1\}^2$ and $Z=XY$. Then any two random variables from $X$, $Y$ and $Z$ are independent and $\mathrm{cov}(XY,Z)=1$.