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 1d comment Discrete Random Variable case All done. What now? 1d comment Let $u=u(x,y)$ be a differentiable function such that $u(x,x^{2}) = 1$ and $u_{x} (x,x^{2}) = x$. Find $u_{y} (x,x^{2})$ Fix some $x\ne0$, the goal is to find some $v$ such that $u(x,x^2+h)=u(x,x^2)+hv+o(h)$ when $h\to0$, then $v=u_y(x,x^2)$. For $|h|$ small enough, $x^2+h>0$ hence $x^2+h=z^2$ with $z=x+\frac{h}{2x}+o(h)$ hence $x=z-\frac{h}{2z}+o(h)$ and $$u(x,x^2+h)=u\left(z-\frac{h}{2z}+o(h),z^2\right)=u(z,z^2)-\frac{h}{2z}u_x(z,{}‌​{}{}{}z^2)+o(h)=1-\frac{h}{2z}z+o(h),$$ that is, $u(x,x^2+h)=u(x,x^2)-\frac{h}2+o(h)$, qed. 1d comment Calculate $P(N_1 = 2 \mid N_3 = 10)$ No, $P(N_1 = 2 | N_1 + (N_3 - N_1) = 10)$ is not $P(N_1 = 2 | N_3 - N_1 = 10 -2)$. To compute $P(N_1 = 2 | N_3 = 10)$, compute $P(N_1=2,N_3=10)=P(N_1=2)P(N_3-N_1=8)$ and $P(N_3=10)$. 2d comment On the linear combination of $\pm 1$ random variables Ach so... finally! Now that every $a_j$ is an integer, $X$ is integer valued hence one can use the identity $$2\pi P(X=0)=\int_0^{2\pi}E(e^{itX})dt=\int_0^{2\pi}\prod_jE(e^{ita_jX_j})dt=\ldots$$ Can you end this? 2d comment Why does $1+2+3+\cdots = -\frac{1}{12}$? Re the three videos linked to in the comments above, the one by MrYouMath is to be commended while the first Numberphile video is, for its main part, a lazy exercise of self complacency, which caused enough reactions to motivate its authors to post the other video linked above, as a kind of (not very convincing, if you ask me) damage control operation. 2d comment Find the value of :$\lim_{\Delta t \rightarrow 0^+} \frac{\epsilon}{\sqrt{16\pi D (\Delta t)^3}}e^{-\epsilon^2/(4D(\Delta t))}$ From the expansion $e^t=1+t+o(t)$ when $t\to0$, naturally. 2d comment Show two norms are equivalent Indeed. Do not hesitate to post a solution to your own question, this practice is actually recommended. 2d comment Periodic orbits of a dynamical system Did you try to look for the keywords I indicated? The periods of this system are well-known... 2d comment Integrals with erf^N OK, the formula in the post and in the link are different... 2d comment Integrals with erf^N Sure that erf(-(x-z)^2/(2σ^2 )) is involved, not simply erf(z)? 2d comment Calculating the covariance matrix And you shouldn't have. If you try to bypass the rules of the site and if this has consequences, do not complain afterwards, right? 2d comment Show two norms are equivalent Sorry but you should be more careful: I pretend that I can find some nonzero $a$ such that $a\leqslant n$ for every $n$ simultaneously. Can you? Then $a$ is not a problem, right? Now, turn to $b$... 2d comment Calculating the covariance matrix You already posted this not long ago, with as much personal input as here (meaning, zero), and you are probably reposting it to circumvent the closure of the first version. Another option would be to add your thoughts on the question ("I am stuck" and "the question is pretty vague to me" and "I don't really understand" do not count). Weren't you given indications on the first installment? 2d comment Set of rational numbers bounded between two irrationals is a closed set? Except that now, "Consider the metric space $\mathbb{R}$ equipped with the standard distance metric" and "Prove that $S$ is closed in the set of rational numbers $\mathbb{Q}$" are contradictory. If the ambient space is $X$ and if one is given $S\subset Y\subset X$, please explain what you mean by "$S$ is closed in $Y$". 2d comment How do we know that $c|a$ if $c=\gcd(a,b)$ I liked the first version of the question better... 2d revised Show two norms are equivalent added 12 characters in body 2d comment Show two norms are equivalent Again lost in what-ifs... Now that you have each $N(e_n)$ and each $\|e_n\|_3$, what can you say about $a$ and $b$ such that $aN(e_n)\leqslant\|e_n\|_3\leqslant bN(e_n)$ for some $n$? For every $n$ simultaneously? (See how I am merely copying parts of your question...) 2d comment need help with abstract algebra You say you need help but you are effectively preventing us to give help... 2d comment Show two norms are equivalent Did you compute $N(e_n)$ and $\|e_n\|_3$? You know, before asking what-ifs and being unsure... 2d comment Finding event with specific probability. Objective is to find the event. @Vincent $q=1-p$.