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Convergence in Distribution in Order Statistics Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. (a) Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$, where $Y_1$ and $Y_n$ represent the first and last order statistics (among $n$ observations) respectively. Answer: I have obtained the CDF of X to be: $$F_x(x) = 1 - e^{-\frac{x}{\theta}}$$ the CDF of $Y_1$ to be: $$F_{Y_1} = 1 = e^{-\frac{xn}{\theta}}$$ the CDF of $M_n$ to be: $$F_{M_n}(x) = \left[1 - \frac{1}{n}e^{-\frac{x}{\theta}}\right]^n$$ with limiting distribution $$e^{-e^{-\frac{x}{\theta}}}$$ the CDF of $T_n$ to be: $$F_{T_n}(x) = 1 - e^{-\frac{x}{\theta}}$$ which is also the limiting distribution. (b) Show that $M_n$ and $T_n$ are asymptotically independent. Answer: I've tried finding the joint PDF for ordered statistics by using the equation $$g_{ij}(y_i, y_j) = \frac{n!}{(i - 1)!(j - i - 1)!(n - j)!}[F(y_i)]^{i - 1}[F(y_j) - F(y_i)]^{j - i - 1}[1 - F(y_j)]^{n - j}f(y_i)f(y_j)$$ however, it is a mess based on my CDFs. Any suggestions on a different approach? Non-principal ideal in $K[x,y]$? Let $K$ be a field. Then $K[x,y]$ is not a PID because $x$ is irreducible but the quotient $K[x,y]/(x)$ is isomorphic to $K[y]$, which is not a field.  So there must be an ideal in there which is not principal. What is an example of such an ideal? We have to evaluate the following integral:  $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$  I tried this:  I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$.  But after that I got stuck.  The book where this is taken from gives the following as the answer: $$\ln(1+t)-\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$ What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] Sweave.sty causing error in verbatim environment? Is it possible to to take a screenshot of a webpage with JavaScript and then submit that back to the server?  I'm not so concerned with browser security issues. etc. as the implementation would be for . But is it possible? Stripped out bottom bracket Instantiating object of type parameter How can I add a class to a block with code (development) For theming reasons I'd like to add a specific class at the "block" level to some of the blocks I'm generating with a custom module. I know there is a module , but I would rather like to do it directly in code (there is no need to change is afterwards).   I also know of , but that does not seem to allow adding or editing classes.  Is it possible to add a class with to a block from within my module code? I'm using Drupal 7. I know you can create a rounded-corner rectangle; but, how do you create a triangle with rounded corners in Photoshop?  I am actually interested in making it from scratch, not basing it on a custom shape that is available in Photoshop. Thanks. What's the "official" explanation for Harry's (and everyone else's) absence from muggle school? What is the difference between '/' and '//' when used for division? Is there a benefit to using one over the other? In Python 2, they both seem to return the same results:  &gt;&gt;&gt; 6/3 2 &gt;&gt;&gt; 6//3 2 Ubuntu 20.04 no sound out of Bluetooth headphones UMVUE of $e^{-\lambda}$ from poisson distribution Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $X_i=0,$ $Y_i=0$ otherwise. Clearly (I checked!) each $Y_i$ and hence $\overline{Y}_n$ is unbiased for $e^{-\lambda}$. Then I should be able to use the Lehmann Scheffe theorem to deduce the UMVUE. The estimator I get looks like $$\frac{1}{n} E\left\{\#X_i:X_i=0 \mid \sum_{i=1}^n X_i\right\}$$ whereas the correct answer is $$\left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i}$$ I'd appreciate if anyone lets me know where I'm going wrong and guides me into the right path. Thank you. Android: Overlay on Android Camera Preview Books for Condensed Matter after Ashcroft/Mermin What are some good condensed matter physics books that can fill the gap between Ashcroft &amp; Mermin and research papers? Suggestions for any specialized topics (such as superconductivity, CFT, topological insulators) are welcomed. How to mount an image file without root permission? Should we avoid negative words when writing error messages? What is the angle between those two matrices over $\mathbb{C}$? Let $\left&lt;\;,\;\right&gt;$ be an inner product defined by $\left&lt;A,B\right&gt;=$ tr$\left(A\overline{B}^{T}\right)$ over $M_n\left(\mathbb{C}\right).$  I am looking for the angle between: $$A=\begin{pmatrix}2+i &amp; 1\\-1&amp;0\end{pmatrix}\quad B=\begin{pmatrix}3-2i &amp; 1+i\\0 &amp; -i\end{pmatrix}$$  First we can see that:$$\overline{A}=\begin{pmatrix}2-i &amp; 1\\-1&amp;0\end{pmatrix}\quad \overline{B}=\begin{pmatrix}3+2i &amp; 1-i\\0 &amp; i\end{pmatrix}$$  $$\Downarrow$$  $$\overline{A}^T=\begin{pmatrix}2-i &amp; -1\\1&amp;0\end{pmatrix}\quad \overline{B}^T=\begin{pmatrix}3+2i &amp; 0\\1-i &amp; i\end{pmatrix}$$  And therefore:   $\left&lt;A,B\right&gt;=$ tr$\left(A\overline{B}^{T}\right)=\left(2+i\right)\left(3+2i\right)+1-i=5+6i$ $\left&lt;A,A\right&gt;=\left(2+i\right)\left(2-i\right)+1+1=7$ $\left&lt;B,B\right&gt;=\left(3-2i\right)\left(3+2i\right)+\left(1+i\right)\left(1-i\right)+1=16$   Now as the angle between any two vectors is defined we get that: $$\cos{\alpha}=\frac{\left&lt;A,B\right&gt;}{\Vert{A}\Vert\Vert{B}\Vert}=\frac{5+6i}{\sqrt{112}}$$  $$\Downarrow$$  $$\alpha = \;?$$  I didn't even know a complex number could be an image for the cosine function.  Thanks for any help. How many MB/sec can I expect copying to USB 3.0 enclosure with SATA drive?