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 Feb5 comment Where is the mistake in this argument about number of different balls with replacement Thanks for the good write-up! I've got a better intuition of the situation now. Feb5 comment Where is the mistake in this argument about number of different balls with replacement Thanks for the answer. Yes that is my intuition too. However I fail to see how the events $D>0$ and $E_k$ are different. If $D>0$, if and only if there exists a $k$ such that $E_k$ is true. Jan30 comment Choosing the vector that minimizes this sum related to the rearrangement inequality @PeterKošinár: I modified the statement slightly following our discussion and my experiments, and I think it is more correct now. Jan28 comment Choosing the vector that minimizes this sum related to the rearrangement inequality @PeterKošinár: Yes it seems there are problems with small $n$. My intention is to work with big values of $n$, but this example is degenerated (but correct and helpful). The intuition here is that it is preferable, in order to minimize the sum, to pair big $x$'s with small $y$'s, in the same line of thought as the rearrangement inequality. In this case the sum cuts short too early for this strategy to prove useful. Perhaps the conjecture is only true for $n\to\infty$? (or perhaps is it not true at all) Jan26 comment Choosing the vector that minimizes this sum related to the rearrangement inequality Thank you, I corrected the example. Regarding your example, we have $S([0]) = 1 < S([1]) = 2$, and the conjectured condition proposes indeed $b^* = [0]$. Jan21 comment Optimal Strategy for Chosing Lottery Tickets @Lost1: It's not just the distribution. In fact I think (still working it out) that we could formulate it this way: the cost always remains the same, and each time you choose ticket A, the next ticket A will have its probability multiplied by a constant factor $k_A<1$, and ticket B remains the same (and vice-versa). But it might be more or less complicated, I need to think about it a bit more. At any rate, thanks for the answer. Jan21 comment Optimal Strategy for Chosing Lottery Tickets @Lost1: I understand your concern, but the thing is that I may have been overly liberate with the assumptions. In fact I do know something about future $c$ and $p$, I just thought initially that it did not matter. Obviously, it does. Jan21 comment Optimal Strategy for Chosing Lottery Tickets @Lost1,Andrey: good point, it is true that with no information on the subsequent deals, something like this might come up. I probably need to rethink my question. Should I update this post ? Dec3 comment Why do XOR and other operators on binary variables qualify as linear? Ok I think I got the idea. But so, why are permutations over $\mathbb{F}_2^n$ considered linear ? For instance, consider the data-dependent rotation $Rot(x,y):\mathbb{F}_2^n \times \mathbb{Z}_n \mapsto \mathbb{F}_2^n$ which consists in a circular left shift of $x$ by $y$ positions. Is it linear in $x$ ? Dec3 comment Why do XOR and other operators on binary variables qualify as linear? @JyrkiLahtonen: Thanks for the comment. I know that XOR is addition mod 2, and this is why I understand it intuitively, but can you give me a mathematical reasonning ? And what do you mean by "takes sums to sums" ? do you refer to the fact that $f(x+y)=f(x)+f(y)$? In this case I don't know how that would translate for binary operators? Jan31 comment Recurrence equation similar to a geometric progression @leonbloy no, I meant the initial conditions ($T(1) = A$ and $T(n) = B$). Nov30 comment Distribution probability of elements and pair-wise differences in a sorted list Indeed, thanks for this great answer :-) Nov30 comment Distribution probability of elements and pair-wise differences in a sorted list Thanks for the answer! I actually had that after I posted the question, but I was not able to check that $\sum_{i=0}^{N-1} \Pr(x_i = x) = 1$, and somehow started doubting on it. The way you put it makes sense though, and I did that sum numerically which seems to be right! A note on $d$: it seems that $\Pr(d_i = 0) = \frac m {N-m+1} > 0$ which makes no sense if elements are unique... Yet you did take that into account in the pdf of $x_i$. Any idea ? Sep19 comment Evaluating a recurrence relation with non-contiguous initial conditions Thank you for your answer. However, I am not looking for solving the recurrence, but rather evaluating the $T(i)$. Sep4 comment Recurrence equation similar to a geometric progression Just a regular bracket. I edited to avoid confusion.