Suppose that $X$ and $Y$ are independent random variables. I wish to prove that $X$ and $Z = Y^2$ are also independent.  I know from the definition of independence that $P({X \in A}$ and $Y \in B$) = $P({X \in A})$$P({Y \in B})$, but I'm not sure how to apply this to the above problem.
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.  If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$ = $f_1(X_1)$ and $Y_2$ = $f_2(X_2)$.  Intuitively, $Y_1$ and $Y_2$ should be independent, and I can't find a counter example, but I am not sure. Could anyone tell me whether they are independent? Does it depend on some properties of $f_1$ and $f_2$?  Thank you.
Remove Background in Inkscape Drawbacks of making public which journal a manuscript is submitted to, while it is under review? Many researchers list out their submitted manuscripts on their website. Some specify the journal name, like     Smith J, Oliver J (under review) My awesome paper. Nature.   while others don't, like     Smith J, Oliver J (under review) My awesome paper.   One reason I can think of as to why some don't mention the journal name is that they may get embarrassed in the case where the current journal rejects the paper. Then, the "circle" will know this paper gets rejected from Nature first and then ends up in an inferior journal.  Besides this, are there any other considerations preventing people from mentioning to which journal they submit a paper to?  P.S.: I am asking because my advisor is a researcher who only mentions accepted  papers on his webpage. As a student hunting jobs, I feel it is advantageous for me to mention my submitted papers on my webpage. I hesitate to do so, as I worry I may make my advisor unhappy by announcing in which journal our papers are currently under review. How can they see Harry's invisibility cloak when they are not wearing it? htaccess subdomain How do I set up an email server? How can I best describe "making a new discovery"? How do I uninstall Ubuntu? How do I format drive and unistall Ubuntu 11.10?  I wanted a dual boot and was given an option to overwrite Windows XP partition and said yes.  Format options don't work.  I tried a few dos prompts and probably not using correct fdisk.  Format options state drive is busy closed all windows.  Tried to boot windows XP and disk says its corrupt. Short story about dead soldiers bodies being reused Should I use a categorical cross-entropy or binary cross-entropy loss for binary predictions? First of all, I realized if I need to perform binary predictions, I have to create at least two classes through performing a one-hot-encoding. Is this correct? However, is binary cross-entropy only for predictions with only one class? If I were to use a categorical cross-entropy loss, which is typically found in most libraries (like TensorFlow), would there be a significant difference?  In fact, what are the exact differences between a categorical and binary cross-entropy? I have never seen an implementation of binary cross-entropy in TensorFlow, so I thought perhaps the categorical one works just as fine. Every finite commutative ring with no zero divisors contains a multiplicative identity? When an object is moving in a circle, won't the frictional force oppose it's motion? And if the velocity is always tangential to the path, there is no component of it towards the center, so there is no component of friction towards the center (since the frictional force is antiparallel). So how can friction be a centripetal force, like in the case of a car rounding a corner? List all the directories under the current one Difference between systemctl and service commands systemd gives us the systemctl commands suite which is mostly used to enable services to start at boot time. We can also start, stop, reload, restart and check status of services with the help of systemctl.  We can do, for example, sudo systemctl enable service_name, and service_name will automatically start at boot time. We can also disable services not to start at boot time.  Is the only difference between the service and systemctl commands that systemctl can be used to enable the start of services at run time? Can we use systemctl on any service? What other significant differences are there? Why my differential equations become True? Why two inverters cascaded in multiplexer control signal terminals? Why cascaded inverters are used in the multiplexer when it is possible to realize the circuit using a single inverter. Is it okay to use my US visa after a name change? I was going to make a filter for the &quot;&quot; by excluding some sites from it.  I noticed  which talks about &quot;negative filtering&quot; but does not explain how one can do it or if it is possible.  There does not seem to be a way to do this, though.  On the filter page, my only options are:  So how can I do &quot;All sites except [list]&quot;? It would also be nice if this could include similar functionality with the meta checkbox.  Something like &quot;All main &amp; meta sites except...&quot; I put a video in my pdf with Acrobat and there I can watch my video. But if I open the pdf on my mobile, I can't watch the video, I only see an image. Is there a way to watch videos of a pdf in a mobile? I have a samsung J5. Edited vs emitted - why one t vs two t's Why does "edited" have one "t" and  and "emitted" has two? "Edit" and "emit" are so similar in spelling and pronunciation.  I keep wanting to type "editted" for some reason. How do I tell Maven to use the latest version of a dependency? In Maven, dependencies are usually set up like this:  &lt;dependency&gt;   &lt;groupId&gt;wonderful-inc&lt;/groupId&gt;   &lt;artifactId&gt;dream-library&lt;/artifactId&gt;   &lt;version&gt;1.2.3&lt;/version&gt; &lt;/dependency&gt;   Now, if you are working with libraries that have frequent releases, constantly updating the &lt;version&gt; tag can be somewhat annoying. Is there any way to tell Maven to always use the latest available version (from the repository)?