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 Dec 18 comment Mean of the deviations from the mean @ronn you have $n+1$ values, count them: $y+0a, y+1a, \ldots, y+na$... Dec 18 comment Integrate $4x/(x^4-1)$ dx @Dr.MV fixed, thanks Dec 16 comment What is the name of the theorem? @AyleanClaraGrandieur see update with full text & hyperlink Dec 15 comment Integral of the product of a power function and an arbitrary exponentiated function not sure it helps but substituting $u = t^a$ yields $$\frac{1}{a} \int \exp\left( \frac{u^{1/a} \ln 2}{\lambda} - \frac{u}{b}\right) du$$ Dec 15 comment Probability - Limits of integration for Z=X+Y, with bivariate density f(x,y) you should have $$f_Z(z) = \int_\mathbb{R} f(x,z-x) dx$$ Dec 15 comment Extending relation to be transitive Say, it is transitive and you are using the matrix representation. Pick some $a \sim b$ and $b \sim c$ and mark them in your matrix. What entry does $a \sim c$ represent? can you generalize this? Dec 15 comment System of differential equation @Ergo your approach isn't always correct. E.g. here, it won't work. Definition of $e^{tA}$ (it is also a matrix) is from the Taylor series in the answer. You can simplifyit greatly by just computing $A^2$. What do you get? Then multiply by $\vec{F}(0)$ on the right to get the answer... Dec 15 comment Distribution of ratio of functions of random variables @charn are you sure the numerator is $\sin(2x)$ and not $\sin^2(x)$? Your function $a(\cdot)$ is not even invertible... Dec 15 comment System of differential equation @Ergo can you please be more specific? what do you not understand exactly? Do you know what $e^A$ means when $A$ is a matrix? Dec 15 comment System of differential equation @Ian thanks, fixed Dec 15 comment Integrate $\displaystyle \int{\frac{x}{1+x^4}}dx$. +1 but i would let the OP do at least some of the work Dec 15 comment Distribution of ratio of functions of random variables @charn you can invert numerically. Doing this analytically is problematic Dec 14 comment If $S = \min\{k \geq 1: X_k - Y_k > \sqrt{\log k}\}$ and $X_k$ and $Y_k$ are iid Normals, is it true that $P(X < \infty) = 1$? I don't see how $_i$ and $Y_i$ depend on $S$? Dec 14 comment Calculating $\lim_\limits{n\to \infty}\frac{n^{2}}{\left(2+\large\frac{1}{n}\right)^{n}}$ @kamil09875 this is a hint, not an answer -- the OP is invited to think about his problem himself. But if you ensist, one transforms $$\frac{n^2}{(2+1/n)^n} = \frac{n^2}{2^n} \left(\frac{2}{2+1/n}\right)^n$$ and bounds the right term by $1$ Dec 14 comment weighted graph problem Please explain clearly. What is the shortest path tree -- it is either a path or a tree? MST will contain the root by definition. Also what makes your minimal actually minimal? Do you mean minimal or minimum? Dec 11 comment unique fixed point problem if it's differentiable (not derivable), it must be continuous. no need to assume more Dec 11 comment Relations on groups. @ThéodoreRozencwajg transitivity means $a\sim b$ and $b \sim c$ imply $a\sim c$. So assume $a \sim b$ and $b \sim c$. So $b^{-1} a \in H$ and $c^{-1}b \ in H$. What can you say about $c^{-1}a$ -- is it in $H$ or not? Dec 10 comment Permutations and Combinations - Disc101 is the order they are walking into the doors important or not? Dec 8 comment How to prove the recurrence relation for this generating function problem? @hlyates Leaving out the factor of $2$, note that $$\frac{1}{1-x} - 1 = \frac{x}{1-x}...$$ The logic is simple $$\sum_{k=1}^\infty x^k = \sum_{k=0}^\infty x^k - x^0 = \frac{1}{1-x} - 1$$ Dec 7 comment Solve a inequality for values of $r$ and $\theta$ You mean to say you need to solve the inequality $LHS > 0$ for the valid region for $r$ in terms of $b$ and $\theta$?