Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed).  Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or similar concepts?  Do we know that $\max(X,Y)$ is normally distributed, maybe a formula for mean and standard deviation in terms of those for $X$ and $Y$?  I checked the usual places (wikipedia, google) but didn't find anything.
Let $X$ be a beta distributed variable with parameters $a$, $b$. Let $Y$ be a beta distributed variable with parameters $c$, $d$. Let $Z = \max(X, Y)$.  Does anybody know of a fast way to compute the mean and variance of $Z$ given $(a, b, c, d)$?  $X$, $Y$ don't necessarily need to have beta distributions, but they should be similar -- in $[0, 1]$.  What I really want to find is a way to quickly approximate the distribution of the sample max of a bunch of random variables. Let $Z = \max(\{X_1, X_2,\dots , X_n\})$. If I know parameters (not necessarily beta-distribution) for $X_1, X_2,\dots, X_n$, can I quickly compute a set of parameters (same distribution as $X_i$) to form a distribution that closely approximates $Z$?    EDIT: To clarify, the purpose of this has to do with programming AI for board games. If we are evaluating a position in a chess game and we determine one move leads to a 60% chance of winning while all others have a 20% chance of winning, then we value the position as a 60% win. This is the minimax algorithm... However, what I'm curious about is whether this can be improved upon. In the simple case, we approximate: $$ \mu_c = \max(\{\mu_1, \mu_2,\dots, \mu_n\}) $$ where $\mu_c$ is the expectation from the current position and $\mu_i$ is the expectation from each of the sub-trees.  It seems like we should be able to do better than this. Can we find a function f that replaces max and closely approximates parameters for the distribution of the sample maximum? $$ (\mu_c, \sigma_c) = f\left(\{(\mu_1,\sigma_1), (\mu_2,\sigma_2), \dots, (\mu_n, \sigma_n)\}\right) $$ In this case, we still want $\mu_i$ to be the expectation from each of the nodes. However, $\sigma_i$ does not necessarily need to be variance/standard deviation; just a parameter that represents uncertainty.  Just as: $$ \max(\{a, b, c\}) = \max\left(\{\max(\{a, b\}), c\}\right) $$ We should expect: $$ f\left(\{a, b, c\}\right) \approx f\left(\{f(\{a, b\}), c\}\right) $$ So $f$ does not necessarily need to take parameters for more than two nodes.  One thing, however, is that $f$ should be relatively fast to compute. If the integrals must be calculated, I think doing double exponential integration might work well, but I'm not sure --
Where to start with Oracle? Hamiltonian Cycle Problem At the moment I'm trying to prove the statement:     $K_n$ is an edge disjoint union of Hamiltonian cycles when $n$ is odd.   ($K_n$ is the complete graph with $n$ vertices)  So far, I think I've come up with a proof. We know the total number of edges in $K_n$ is $n(n-1)/2$ (or $n \choose 2$) and we can split our graph into individual Hamiltonian cycles of degree 2.   We also know that for n vertices all having degree 2, there must consequently be $n$ edges. Thus we write $n(n-1)/2 = n + n + ... n$ (here I'm just splitting $K_n$'s edges into some number of distinct Hamiltonian paths) and the deduction that $n$ must be odd follows easily.  However, the assumption I made - that we can always split $K_n$ into Hamiltonian paths of degree 2 if $K_n$ can be written as a disjoint union described above - I'm having trouble proving. I've only relied on trying different values for $n$ trials and it hasn't faltered yet. So, I'm asking:     If it is true, how do you prove that if $K_n$ can be split into distinct Hamiltonian cycles, it can be split in such a way that each Hamiltonian cycle is of degree 2? Find & Delete duplicated files on multiple harddisks at once I have 4 hard disks and want to find out which files on this four harddisks (including sub directories) are duplicates. It should be checked not only within a harddisk but against all the others as well.  The hard disks are large (3TB) therefor it has to be efficient (first filename than checksum checks or so) How to move an object's origin to the center of its bounding box? What I want: I want to set the origin of an object to the center of its bounding box using a python command.  What I've tried:  bpy.ops.object.origin_set(type='ORIGIN_CENTER_OF_MASS')   and  bpy.ops.object.origin_set(type='ORIGIN_GEOMETRY')   What happened:     As you can see, the origin isn't centered to the bounding box.  EDIT: I asked the same question as  with the addition of setting the origin to this center. My input file is,  ami21 ami65 ami67 ami66 ami88 ami76 ami29 ami55  ami54 ami32   Using a single command line I need the output as,  ami22 ami66 ami68 ami67 ami89 ami77 ami30 ami56 ami55 ami33   I used the command  awk -vRS=ami '{$0=$0+1;ORS=RT}++n' inputfile &gt; outputfile   but I got the outputfile as ami21ami65ami67ami66ami88ami76ami29ami55ami54ami32  i.e. all strings are written in same line and without space. Could anyone suggext me some better command line. Is there a single word for a "friend of a friend"? I am looking for a word that I could use instead of "friend of a friend". On certain occasions, it happens that you may want to talk about a friend of a friend with someone else.   In such cases how would you refer to this friend? Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\right)^2}{n-1}?$$ failed to get D-Bus connection: Operation not permitted I'm trying to list services on my CentOS image running in Docker using  systemctl list-units     but I get this error message:  Failed to get D-Bus connection: Operation not permitted   Any suggestions what the problem might be? Do the topologies with the same convergent sequences coincide? Do the same sequential convergence in two topologies the result are the same topologies? Let $X$ be topological  space with topologies $\tau$, $\tau'$. Let $(x_{n})\in X$ and $x\in X$ and $(x_{n})\rightarrow x$ with $\tau$ if and only if $(x_{n})\rightarrow x$ with $\tau'$. Do the two topologies on $X$ coincide? How to search entire hard drive for a file? Voltage Controlled Current Source integrated circuit? Exists? Does anyone know of a Voltage Controlled Current Source IC? Or do you know how I can build one with as few other components as possible taking up a minimal amount of space?  current range needed is from 0mA to 350mA Script access to tangent and bitangent, per face: how? Suppose you have a mesh; you go into Edit mode,  set Transform Orientation to Normal,  show 3D manipulator widget, Pivot Point = Active Element, face select mode, and then you select any one face: immediately you will see a blue arrow representing the face normal, and also a red and a green arrow representing the face's tangent and bitangent.  What is the most straightforward way to refer  to said tangent and bitangent vectors from a Python script? The API documentation confuses me with loops and uv's, yet, clearly, Blender assigns a tangent and bitangent to each face without hesitation! Related questions in "Ask Question" page is better than the search tool Escape Key in the Chat How to prove this inequality $ x + \frac{1}{x} \geq 2 $ I was asked to prove that:   $$x + \frac{1}{x}\geqslant 2$$  for all values of $ x &gt; 0 $  I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a feeling that this is an unprofessional way of proving this. So how do I prove this inequality? Let $f : \mathbb R^n \rightarrow \mathbb R$ be a measurable function.  We know that $f$ is not necessarily locally integrable - but how hard can a measurable $f$ fail to be so? For example, is it true that $f$ is locally integrable outside of a negligible set $N \subset \mathbb R^n$?   Edit: In consideration of the comment by zhs, let me correct the phrasing of the question. I was originally wondering whether the 'singular' behaviour of a measurable function can be localized to a 'small' set.   A motivational example is $f(x) = 1/x$ over the real line. It is locally integrable in the complement of the origin. Hence, a conceivable phenomenon, which I had in mind originally when asking the question, could be that a measurable $f$ is integrable over open sets that are compactly contained in the complement of some negligible sets. VirtualBox - how to increase video memory? Determine the probability densities for coefficients of equation Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?