Define $X$ and $Y$ to be two exponential random variables. The probability distribution functions are $f_{X}(x)=\lambda e^{-\lambda x}$ and $f_{Y}(y)=\delta e^{-\delta y}$. Note that $X$ and $Y$ are independent.  Define $Z=X +\frac{a Y}{Y+b}$, where $a$ and $b$ are some positive constants. My goal is to derive the CDF of $Z$, i.e. I want to calculate $P(Z &lt;z)$.  My approach consists in deriving the CDF of $R=\frac{a Y}{Y+b}$, and then trying to find the CDF of the sum, which will be a function of the CDF of $R$. So I have started as follows: $P(\frac{a Y}{Y+b} &lt; r)=P(Y(a-r) &lt;rb)$. It is clear that here $a$ should be $&gt;r$, because otherwise we get something negative.. Thus, following this approach forces us to assume that $a&gt;r$, something that I don't know if I can assume or not.    Question: is this assumption necessary to be able to derive the CDF? is there any other approach that I can adopt to derive the CDF without making this assumption?
Let $X_1$ and $X_2$ be two independent exponential random variables with the PDFs $f_{X_1}(x_1)=\lambda_1 \exp(-\lambda_1 x_1)$ and $f_{X_2}(x_2)=\lambda_2 \exp(-\lambda_2 x_2)$.   Let $X= \frac{X_1X_2}{X_1+X_2+a}$ be a r.v. for which I want to derive the CDF, where $a$ is a positive constant.  In other words, I need to calculate $\mathbb{P}\{X &lt;x \}$.  Any ideas or hints?  Edit (attempt):  Using a similar approach to that used , we have $X_1 \in [0, \infty)$ and  $X_2 \in [0, \frac{x(x_1+a)}{x_1-x})$. The latter interval results from: $\frac{x_1x_2}{x_1+x_2+a} &lt; x$, thus $x_2(x_1-x)&lt;x(x_1+a)$ which implies $x_2 &lt; \frac{x(x_1+a)}{x_1-x}$; but I am not sure if this is correct since $x_1-x$ can sometimes be negative (?). Based on the above, we can write: \begin{align}  \mathbb{P} \{ \frac{X_1X_2}{X_1+X_2+a} &lt; x \}  &amp; = \int_{x_1=0}^\infty \int_{x_2=0}^{ \frac{x(x_1+a)}{x_1-x}} \lambda_1 \exp(-\lambda_1 x_1) \lambda_2 \exp(-\lambda_2 x_2)  dx_2 dx_1  \\ &amp;= \int_{x_1=0}^\infty \left(1-\exp(-\lambda_2\frac{x(x_1+a)}{x_1-x}) \right) \lambda_1 \exp(-\lambda_1 x_1) dx_1  \\ &amp; = 1-\lambda_1 \int_{x_1=x}^\infty \exp(-\lambda_2\frac{x(x_1+a)}{x_1-x} ) \exp(-\lambda_1 x_1)   dx_1 \end{align} In the last equality: for the integral of $\lambda_1 \exp(-\lambda_1 x_1) dx_1$, I suppose that $x_1 \in [0,\infty)$, whereas for the second term I suppose that $x_1 \ge x$ (since $x_2$ should be positive) (?).  Is my approach correct ?
Can i prevent/restrict standard user from installing/uninstalling software in Ubuntu 12.04 desktop? Why is Leia okay with being tracked to Yavin? In A New Hope, after Han and Luke fight off the TIE Fighters while escaping from the Death Star, these exchanges happen:             INT. DEATH STAR - CONTROL ROOM             Darth Vader strides into the control room, where Tarkin is             watching the huge view screen. A sea of stars is before him.                                   TARKIN                      Are they away?                                   VADER                      They have just made the jump into                       hyperspace.                                   TARKIN                      You're sure the homing beacon is                       secure aboard their ship? I'm taking                       an awful risk, Vader. This had better                       work.             INT. MILLENNIUM FALCON - COCKPIT             Han, removes his gloves and smiling, is at the controls of             the ship. Chewie moves into the aft section to check the             damage. Leia is seated near Han.                                   HAN                      Not a bad bit of rescuing, huh? You                       know, sometimes I even amaze myself.                                   LEIA                      That doesn't sound too hard. Besides,                       they let us go. It's the only                       explanation for the ease of our                       escape.                                   HAN                      Easy... you call that easy?                                   LEIA                      They're tracking us!                                   HAN                      Not this ship, sister.             Frustrated, Leia shakes her head.                                   LEIA                      At least the information in Artoo is                       still intact.                                   HAN                      What's so important? What's he                       carrying?                                   LEIA                      The technical readouts of that battle                       station. I only hope that when the                       data is analyzed, a weakness can be                       found. It's not over yet!   Leia doesn't even know if there is a weakness, let alone whether they'll find one. If she (correctly) realizes that they're being tracked, why doesn't she suggest that the Millennium Falcon go somewhere besides Yavin 4 instead of just frustratedly shaking her head? I know the plans , but it seems unlikely that continuing to fly to the secret Rebel base in the ship she's certain is being tracked is the only option. They are already in hyperspace when this conversation happens, but they apparently have time to talk for a little while after, so presumably they'd have time to stop and reroute. In QGIS I would like to clip a raster precipitation layer using an admin boundary layer that is vector data. However the Geoprocessing tools seem to be usable only for vector data. How can I clip this precipitation layer? What is the difference between “say” and “tell”? I really get confused when to use say and when to use tell. Which is appropriate in the following:   What did he tell? / What did he say? What are you saying? / What are you telling? Regular Expression: Numeric range A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{&gt;0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} , \times )$, not required to be positive, one can argue the group contains elements with order 2 (namely all negatives). In the case of the requirement for all rationals to be positive this argument does not fly. What trivial fact am I missing here? Next month I am traveling to country X, where I would need Y minutes of calls, Z text messages and most importantly Q megabytes of data. How can I find the cheapest/most reliable SIM card for my needs? How to work with iAds using xcode 3.1.3 Recently I've installed Kubuntu in a newly bought laptop with an Intel WiFi 6 AX200 network card, I can connect to the wifi without apparent problems but when I open the driver manager in the settings I face this weird situation:  Intel Corporation: Wi-Fi 6 AX200 This device is not working.  [ ] Using iwlwifi driver backport in DKMS format from backport-iwlwifi-dkms (open source)  [ ] Continue using a manually installed driver  [x] Do not use the device   Note that "Continue using a manually installed driver" is the only voice selectable in the list.  Here is some additional information:  $ lspci -nnk | grep 0280 -A3 02:00.0 Network controller [0280]: Intel Corporation Wi-Fi 6 AX200 [8086:2723] (rev 1a)         Subsystem: Intel Corporation Wi-Fi 6 AX200 [8086:0084]         Kernel driver in use: iwlwifi         Kernel modules: iwlwifi  $ rfkill list all 0: hci0: Bluetooth         Soft blocked: yes         Hard blocked: no 1: phy0: Wireless LAN         Soft blocked: no         Hard blocked: no  $ dmesg | grep iwl [    2.888227] iwlwifi 0000:02:00.0: enabling device (0000 -&gt; 0002) [    2.945179] iwlwifi 0000:02:00.0: Direct firmware load for iwlwifi-cc-a0-50.ucode failed with error -2 [    2.945217] iwlwifi 0000:02:00.0: Direct firmware load for iwlwifi-cc-a0-49.ucode failed with error -2 [    2.947244] iwlwifi 0000:02:00.0: TLV_FW_FSEQ_VERSION: FSEQ Version: 43.2.23.17 [    2.947250] iwlwifi 0000:02:00.0: Found debug destination: EXTERNAL_DRAM [    2.947252] iwlwifi 0000:02:00.0: Found debug configuration: 0 [    2.947697] iwlwifi 0000:02:00.0: loaded firmware version 48.4fa0041f.0 op_mode iwlmvm [    3.145585] iwlwifi 0000:02:00.0: Detected Intel(R) Wi-Fi 6 AX200 160MHz, REV=0x340 [    3.159226] iwlwifi 0000:02:00.0: Applying debug destination EXTERNAL_DRAM [    3.160060] iwlwifi 0000:02:00.0: Allocated 0x00400000 bytes for firmware monitor. [    3.336456] iwlwifi 0000:02:00.0: base HW address: 50:eb:71:d7:0b:40 [    3.353553] iwlwifi 0000:02:00.0 wlp2s0: renamed from wlan0 [    4.457920] iwlwifi 0000:02:00.0: Applying debug destination EXTERNAL_DRAM [    4.621843] iwlwifi 0000:02:00.0: FW already configured (0) - re-configuring [    4.629999] iwlwifi 0000:02:00.0: BIOS contains WGDS but no WRDS   Someone have idea why this happened? How can one rig a Stirling engine? (Piston) I've only done a very little animation and rigging before, and I keep getting stuck trying to rig the pistons and rotor of the Stirling engine I built.     What should happen is that one piston moves in as the other moves out, and through the attached arms, the rotor should spin. Animating the rotor and the piston heads was easy enough, but getting the arms that go from the piston heads to the end of the shaft arm to move right endlessly confuses me.   Each end of each arm moves differently over the course of a cycle. With animating, I can't figure out how to interpolate so the arms stay where they should.  With armatures, I have tried for a while, but keep getting stuck when it comes to how to have one end of a bone moving in a circle while the other end is moving in a straight line.  Edit: The answer below elegantly resolved my problem, so i've updated this with the finished .blend file in case it's useful someone else.   Here is the .blend file: Icon missing when application is launched I've created an application launcher in /usr/share/applications that launches rdesktop. The file specifies an icon name and I've added the icons to:   /usr/share/icons/HighContrast/48x48/apps /usr/share/icons/HighContrast/22x22/apps /usr/share/icons/HighContrast/256x256/apps /usr/share/icons/HighContrast/32x32/apps /usr/share/icons/HighContrast/scalable/apps-extra /usr/share/icons/HighContrast/24x24/apps /usr/share/icons/HighContrast/16x16/apps /usr/share/icons/hicolor/scalable/apps /usr/share/icons/hicolor/128x128/apps /usr/share/icons/hicolor/16x16/apps   I then drug the launcher to the taskbar and the launcher with the correct logo is visible in the taskbar. However, when I start the task, a new icon is created in the taskbar, a square with a question mark in the center.  What icons do I need and where, so that the taskbar displays the correct icon for the application when it is launched?  The .desktop file contents:   [Desktop Entry] Version=1.0 Terminal=false Icon=mswindows Type=Application Categories=Network; Exec=rdesktop win7shared1 -g 1680x1000  Name=win7shared1 GenericName=rdesktop Comment=Open windows desktop  Keywords=Windows StartupNotify=false I'd like to create a Categorized style in QGIS, classify with it data in a shapefile and then save the shapefile with the new categorized style. Is it possible? Allow deleting a tag wiki or excerpt Sometimes a tag wiki excerpt or body ends up with unacceptable or useless content: plagiarized text that must be removed, or a totally unhelpful dictionary definition. (For example:  — and I've seen this situation many times before.)  The only way to get rid of that content is to submit replacement text that meets the minimum length requirement. This is not always easy as writing a decent wiki, and especially a decent excerpt, requires subject knowledge.  Currently the users or even moderators who are alerted to unacceptable content in a tag wiki cannot remove that content without providing their own, which they are not always in a position to do. Yet these wikis would be better off with no content than with either unacceptable or unhelpful content, or some content written by a non-expert in a hurry.  Please provide a way for users to blank a tag wiki excerpt or body. I'm not sure what the reputation requirement should be: same mechanism as for tag wiki edits (suggested edits with approval), or only done directly by a user with the trusted user (20k) privilege, or moderator-only. What language is the Ubuntu Software Center written in? Just been wondering, since elementary AppCenter is written with Vala and Gtk+3.  Also, what GUI wrapper does it use? This is a question in general, not specific to any method or data set. How do we deal with a class imbalance problem in Supervised Machine learning where the number of 0 is around 90% and number of 1 is around 10% in your dataset.How do we optimally train the classifier.  One of the ways which I follow is sampling to make the dataset balanced and then train the classifier and repeat this for multiple samples.   I feel this is random, Is there any framework to approach these kind of problems. Difference between "right of=" and "right=of" in PGF/TikZ In the PGF/TikZ manual, sometimes I see the option right of=somenode instead of right=of somenode. They look very similar, but the effects are different.  The distance between nodes positioned with the latter option is boundary-to-boundary, as stated in the manual. However, with the first option, it seems that the distance is shorter. I couldn't find any explanation of the first option in the manual.  My question is: what is the difference between the two options? Is there any explanation in the manual that I missed? I added addresses of python 2.7, and Anaconda 4.1.1(Python 3.5) to PATH variable in Windows 10, whenever I run "python" from cmd, program which was added earlier is executed.How can I access both of them depending on my choice without altering name of either Water Wars: Time Travel TV Show? Suppose I have a node in the shape of a circle. How can I make its border thicker?   \node[circle, draw=blue!80, inner sep=0pt, minimum size=12pt] (1) at (0,0) {1}; "Open With" dialog ignores my selection I am running Windows 7 x64. I have a file type that I want to associate with Microsoft Access 2003.  I right-click on the file and select Open With->Choose Default program. In the "Open With" dialog, I select the browse button. I choose Program Files (x86)\Microsoft Office\Office11\msaccess.exe and press "Open".  I end up back at the "Open With" dialog, but nothing has changed. Access is not listed as a program to use to open the file with. The original program (Adobe Acrobat) that is associated with this program is still selected.  Any tips on getting this association to work?