This is one of the problem in the Allan Gut's Second Course for Probability.  Let $X_1$, $X_2$ be independent standard gaussian e.g. N(0,1).   Let $Y_1 = \frac{X_1^2 - X_2^2}{\sqrt{X_1^2 + X_2^2}}$, $Y_2 = \frac{2X_1X_2}{\sqrt{X_1^2 + X_2^2}}$. Show $Y_1$, $Y_2$ are independent standard normal.   It is easy to show $Y_1$, $Y_2$ has mean 0, variance 1 and covariance 0. However, I am having trouble show they are joint normal, since the transformation is non-linear.
It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$  Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$  I was thinking of making polar transformations $X=r \cos \theta, Y=r \sin \theta$  Then I am getting stuck at ranges of $\theta$
Get index of each capture in a JavaScript regex Why does this intentionally incorrect use of strcpy not fail horribly? When my phone MOTOROLA G3(2015) is updated to Marshmallow, its not showing files when connected to PC. While connecting, it shows options as charing only, PTP, MTP, MIDI. When I select MTP, it shows the external card, but not showing the contents, where as internal memory is detected and all files are shown. It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.).   If two pieces of the same metal are touched together, why don't they bond?   It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)? Show Console in Windows Application? Reprojected MODIS NDVI has range from -32768 to 32767, expected -1 to 1 Confusion in symbol of subset. If every element of set A ia an element of set B then we say that A is subset of B. A is said to be proper subset if A does not equal B.  My question is - whether the symbol we use to show subset is  same as that for proper subset? I have seen that A is subset of A itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this. I've been writing checks this way for the last ~15 years or so:    Namely, the amount field is written using numbers like     1,500 and 00/100    rather than the tedious, painful spelling out of      One Thousand Five Hundred and 00/100   99 times out of 100, this works fine. But occasionally you'll get certain rare human tellers who simply refuse to believe this is a valid check and you have to fight over it. I still figure the time I've saved by not writing out all those ridiculous numbers-as-words is still worth it, overall, but it's definitely irritating.  So: who's right? Is a check with the amount written in numbers valid ... or not? Is there a free site to play Yu-Gi-Oh? Replace Line Breaks in a String C# There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?  Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.  However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.  This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful! How to extract the last inserted row in SQL Server? When an insert statement is executed one or more rows is inserted into the table, is there any way to extract the last inserted row in SQL Server? Which airlines allow in-cabin pets on transatlantic flights? Interval preserving transformations are linear in special relativity How to save an activity state using save instance state? I've been working on the Android SDK platform, and it is a little unclear how to save an application's state. So given this minor re-tooling of the 'Hello, Android' example:  package com.android.hello;  import android.app.Activity; import android.os.Bundle; import android.widget.TextView;  public class HelloAndroid extends Activity {    private TextView mTextView = null;    /** Called when the activity is first created. */   @Override   public void onCreate(Bundle savedInstanceState) {     super.onCreate(savedInstanceState);      mTextView = new TextView(this);      if (savedInstanceState == null) {        mTextView.setText("Welcome to HelloAndroid!");     } else {        mTextView.setText("Welcome back.");     }      setContentView(mTextView);   } }   I thought it would be enough for the simplest case, but it always responds with the first message, no matter how I navigate away from the app.  I'm sure the solution is as simple as overriding onPause or something like that, but I've been poking away in the documentation for 30 minutes or so and haven't found anything obvious. Why app updates download entire app instead of only "updates"? How do I calculate the challenge rating of monsters that feature abilities that do not directly do damage?  For example, my homebrew monster has the two following abilities:     The target takes a Constitution saving throw. On a failure, it drops to 0   hit points.   This one could be used on a character with 10 hit points, or 100, which creates drastically different damage results depending on character level.     All creatures within 60ft take a Wisdom saving throw. Each target that fails is stunned for 1 round.   This can be extremely dangerous if this monster is paired with monsters that can make use of the stun condition, but it is virtually useless if the monster is alone.  I'm sure there are infinitely more examples of this type of thing.  How do we calculate CR for monsters that do not rely on standard attacks? Why don't you have to press CTRL+ALT+DEL to logon anymore in Windows 8? In previous versions of Windows, you had to press Ctrl + Alt + Delete to logon, or to unlock a locked workstation.  This was because this key sequence was recognized only by the OS and thus other software couldn't intercept it and display a spoofed logon screen to capture passwords.  Starting with Windows 8, you now just have to press Enter to get to the logon screen.  What is to stop someone from writing a fake logon screen?  Did Windows 8 add some sort of new security mechanism to mitigate this security issue? I have two files in these formats:   file1 :  air smell hand dude road cat   file 2:   air,4,21,01,13,3,2 smell,21,4,2,5,6 dude,1,31,42,1 road,1,4,2,1,4 cat,1,5,6,3,1 hand,1,4,2,1,6 mouse,1,3,5,6,2   what I want to do is print the entire row of file 2, if the first string in column 1 of file 2 is found in file 1, and I want to keep the order of file 1.   expected output:   air,4,21,01,13,3,2 smell,21,4,2,5,6 hand,1,4,2,1,6 dude,1,31,42,1 road,1,4,2,1,4 cat,1,5,6,3,1 How does friction stop a car?