Let $f(x)=1/3$ if $-1<x<2$ and zero elsewhere, be the PDF of X. Find the CDF and PDF of $Y=X^2$.
density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution
Round function in Excel, worksheet function vs VBA Find missing indexes tool available to find missing indexes in mssql 2008 ? can we find the missing index using Database Engine Tuning Advisor I absolutely hate it when I watch a video on YouTube and it automatically loads another video page after it's done playing.  This is especially annoying when I'm in the middle of typing a comment as it will cause me to lose what I have typed.  I'd like the video to simply stop and for no further action to take place once it is finished playing.  Is there any way to accomplish this? Replace links with the current question title in comments In questions, links get converted into question titles (see )  Can the same functionality be added to comments? I am technically knowledgeable enough to know that this most likely isn't possible, BUT, I am curious just the same (it must be put to rest). Is there anything preventing us from creating a means to use two different internet connections, like two different ethernet cards or an ethernet card and a wireless card, and using the two IP addresses at once for more internet speed?  My specific situation is that I would like to use both a wireless card and my wired connection on my machine running Kubuntu 13.10. Using two internet connections would allow me to do things like:   Torrent and browse simultaneously without lag Open Firefox with multiple tabs faster (?) Download a couple or a few large files at once double time Increase overall internet connection speed (?) msiexec is command prompt software that installs an  program. But I have found that you can install an MSI file from the command line by just typing in the name of the MSI file on the command line.  But in order to uninstall the MSI file, it seems you have to call the msiexec program and give it a /x or /uninstall.  How can I uninstall an MSI from the command line without using the msiexec routine? Invoking JavaScript code in an iframe from the parent page What is the meaning of modular fractions? What is the meaning of $$ \\\frac{m}{n}\equiv k\pmod l $$ and $$ \\\frac{m}{n} \mod l $$ $m, n, k, l \in\mathbb N$ and $\gcd(m, n)=1$. Running two MySQL servers that use the same data directory Regex pattern to choose data BETWEEN matching quotation marks Functions which are Continuous, but not Bicontinuous What are some examples of functions which are continuous, but whose inverse is not continuous?  nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry. What should I look for when buying chain lube / lubricant? In the past, I personally have only used WD-40.   However, I have occasionally had my bicycle looked over at the locally run, free service sessions and most of the mechanics there use a hand applied type of greasy gel (I forget the name).  Questions:   Is WD-40 suitable for bicycle chains? Would alternatives provide better chain longevity? When shopping around for lubricant, what are the most important things to look for? What do you use and why?  (optional added extra) Any tips for reducing chain wear? How to change keyboard layout via command line cmd.exe on Windows XP/7? Is is possible to change the keyboard layout via command line on Windows XP/7? I have Lubuntu 16.10 installed on my disk mounted as /.  The problem is that the CUPS log file( /var/log/cups/error.log)  grows all the time until I no have free space left on the disk...   So when I delete that file, the space on disk freed up again  I already tried:    fsck disk usage analyzer every apt-cleaning/autocleaning, and dependencies fixing.    How should I proceed?  du -sxh:  9,0G    .  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type      File system     Flags  1      32,3kB  50,3GB  50,3GB  primary   ext4            boot  2      50,3GB  160GB   110GB   extended  5      50,3GB  158GB   108GB   logical   ext4  6      158GB   160GB   2136MB  logical   linux-swap(v1)   parted --list &amp;&amp; sudo df -h:  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type     File system  Flags  1      32,3kB  50,3GB  50,3GB  primary  ext4         boot  2      50,3GB  160GB   110GB   primary  ext4   Filesystem      Size  Used Avail Use% Mounted on udev            985M     0  985M   0% /dev tmpfs           201M  6,3M  195M   4% /run /dev/sda1        46G   43G  709M  99% / tmpfs          1003M  188K 1003M   1% /dev/shm tmpfs           5,0M  4,0K  5,0M   1% /run/lock tmpfs          1003M     0 1003M   0% /sys/fs/cgroup tmpfs           201M   36K  201M   1% /run/user/1000 /dev/sda2       101G   60M   96G   1% /media/aram/ How to import sextante(processing) for QGIS 1.9 (2.0) Variance of sample mean. Let $X_1, X_2, \ldots X_n$ be random variables with mean $\mu$ and variance $\sigma^2$. We define the sample mean $\bar{X} = \sum_{i = 1}^n X_i$.  I am having trouble understanding why the variance of $\bar{X}$ is $\sigma^2 / n$.  Here is what I worked out thus far:  $$\begin{gather} \sigma_\bar{X}^2 = E((\bar{X} - \mu)^2) = E(\bar{X}^2 - 2\bar{X}\mu + \mu^2) = E(\bar{X}^2) - 2\mu E(\bar{X}) + E(\mu^2) = E(\bar{X}^2) - \mu. \end{gather}$$  However, I'm not sure what to do with the $E(\bar{X}^2)$. If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity This is a reworking of a previous question  which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the solutions used the Poisson kernel, which is not the desired solution in the text, because the Poisson kernel has not been introduced yet. The other solutions were for problems that were similar, but whose solutions were only partially applicable, as I understand them.  I'm working on a problem from p. 166 of Lars Ahlfors' Complex Analysis:     If $u$ is harmonic and bounded in $0 &lt; |z| &lt; \rho$, show that the origin is a removable singularity in the sense that $u$ becomes harmonic in $|z|&lt;\rho$ when $u(0)$ is properly defined.   My Solution So Far:   Here's what I've got for a non-Poisson kernel solution: We want to find a harmonic conjugate for $u$ in the punctured disk, and then we can apply big Picard to show that $0$ is a removable singularity. The problem is that the punctured disc is not simply connected.   If we take $\sigma = (u \circ \exp)(z): L \to \mathbb{C}$, where $L$ is the left half-plane, now we have $\sigma$ harmonic, and defined in a simply connected set $L$, so it has a harmonic conjugate $\tau$. Now we wish to take $v = (\tau \circ \log):\{0 &lt; |z| &lt; \rho\}\to \mathbb{C}$, but in order to do so we need $\tau$ to be $2\pi i$-periodic. This is where my proof breaks down.  One of the similar problems which readers referred me to used the following strategy: note that since $\sigma$ is $2\pi i$-periodic, and $\sigma, \tau$ satisfy the Cauchy-Riemann equations, we have that $\nabla\tau(z) = \nabla\tau(z + 2 \pi i)$ for all $z\in L$, or in other words, $\tau(z) - \tau(z+2\pi i)$ is a constant function, say, equal to $\alpha \in \mathbb{R}$. Then, the solution suggests, we can set $$g(z) = u + iv - \frac{c}{2\pi}z,$$  in order to make $g$ $2\pi i$-periodic, and then we can use $(g\circ \log)$ on $\{0 &lt; |z| &lt; \rho\}$. But the reason why this won't work for my purposes is: now we have changed $u$! On the other hand, we can't simply subtract $\frac{c}{2\pi}\Im{(z)}$, and try to leave $u$ untouched, because now $g$ is not analytic. How can we get around this?  Another strategy entirely might be to try to use the mean-value property. Ahlfors discusses that a lot in the preceding section, and his main example is the punctured disc. Since $u$ is harmonic, we know that the average over circles centered at zero is a constant... How to do multiple lines within braces? Can't understand a proof: Let $a,b,c$ be integers. If $a$ and $b$ divide $c$, then $lcm(a,b)$ also divides $c$  Proof I've been shown:  Since $c$ is a common multiple of $a,b$, it means that $lcm(a,b) \leq c$.  So: $c=q\cdot lcm(a,b)+r$, $q \in \mathbb Z$ and $0\leq r \leq lcm(a,b)$  Since $c$ and $lcm(a,b)$ are common multiples of $a,b$, then so is $r$. But $m$ is the least common divisor, then it means $r=0$.   [Proof taken from: http://www.math.utah.edu/~wood/4400/hw1sol.pdf]  I cannot understand the last step... Why does it imply that $r=0$?   Thank you! What is the meaning of this "let there be light" joke?