Assume $X_i$ probability density function is : $$f(x,\lambda)=\Bbb{I}_{(0,\infty)}(x)\lambda \exp(-\lambda x)$$  how to find the probability density function of $\sum X_i$ ? The result is $$\Bbb{I}_{(0,\infty)}(x)\frac{1}{(n-1)!}\lambda^nx^{n-1} \exp(-\lambda x)$$  But I don't know how to find it.  where $\Bbb{I}_{A}(x)=\begin{cases} 1, &amp; x \in A \\ 0, &amp; x\not\in A \end{cases}$
I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function.  Well, I know that $P(T_i&gt;t)=e^{-\lambda t}$ and therefore $f_{T_i}(t)=\lambda e^{-\lambda t}$ so I need to find $P(T_1+\cdots+T_n&gt;t)$ and take the derivative. But I cannot expand the probability term, you have any ideas?
I will soon be attending a Ph.D program in mathematics. I do not know how common this practice is, but the department has a requirement that graduate students must learn a foreign language and be able to read and write mathematics in that language. The three choices are French, German, or Russian. I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career. Thanks in advance in any response. This is a recurring question (see ,  and ), but I have a different spin.  Suppose I have a bunch of samples from a generic MCMC sampler. For each sample $\theta$, I know the value of the log likelihood $\log f(\textbf{x} | \theta)$ and of the log prior $\log f(\theta)$. If it helps, I also know the value of the log likelihood per data point, $\log f(x_i | \theta)$ (this information helps with certain methods, such as WAIC and PSIS-LOO).  I want to obtain a (crude) estimate of the marginal likelihood, just with the samples that I have, and possibly a few other function evaluations (but without rerunning an ad hoc MCMC).  First of all, let's clear the table. We all know that the harmonic estimator is the . Let's move on. If you are doing Gibbs sampling with priors and posteriors in closed form, you can use ; but I am not sure how to generalize outside of those cases. There are also methods that require you to modify the sampling procedure (such as via ), but I am not interested in that here.  The approach I am thinking of consists of approximating the underlying distribution with a parametric (or nonparametric) shape $g(\theta)$, and then figuring out the normalization constant $Z$ as a 1-D optimization problem (i.e., the $Z$ that minimizes some error between $Z g(\theta)$ and $f(\textbf{x}|\theta) f(\theta)$, evaluated on the samples). In the simplest case, suppose that the posterior is roughly multivariate normal, I can fit $g(\theta)$ as a multivariate normal and get something similar to a Laplace approximation (I might want to use a few additional function evaluations to refine the position of the mode). However, I could use as $g(\theta)$ a more flexible family such as a variational mixture of multivariate $t$ distributions.  I appreciate that this method works only if $Z g(\theta)$ is a reasonable approximation to $f(\textbf{x}|\theta) f(\theta)$, but any reason or cautionary tale of why it would be very unwise to do it? Any reading that you would recommend?  The fully nonparametric approach uses some nonparametric family, such as a Gaussian process (GP), to approximate $\log f(\textbf{x}|\theta) + \log f(\theta)$ (or some other nonlinear transformation thereof, such as the square root), and  to implicitly integrate over the underlying target (see  and ). This seems to be an interesting alternative approach, but analogous in spirit (also, note that GPs would be unwieldy in my case). Show the number of up-votes and down-votes on the mobile site theme It seems we can't view the number of up-votes and down-votes a question has from the mobile site. Can this be added or is there a reason it isn't available? If two functionals have the same kernel, then one is a multiple of the other I would like some help with this exercise.  Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.  I expect your suggestions.  Thanks! Fontconfig warning: "/etc/fonts/conf.d/90-fonts-unfonts-core.conf" I am getting the following error frequently. (Example: when running zenity,firefox etc. commands / applications.)  Fontconfig warning: "/etc/fonts/conf.d/90-fonts-unfonts-core.conf", line 11: Having multiple values in &lt;test&gt; isn't supported and may not work as expected   Here is contents of file /etc/fonts/conf.d/90-fonts-unfonts-core.conf:  &lt;?xml version="1.0"?&gt; &lt;!DOCTYPE fontconfig SYSTEM "fonts.dtd"&gt; &lt;fontconfig&gt;     &lt;match target="font"&gt;         &lt;test name="family" qual="any"&gt;             &lt;string&gt;은 바탕&lt;/string&gt;&lt;string&gt;UnBatang&lt;/string&gt;                 &lt;string&gt;은 돋움&lt;/string&gt;&lt;string&gt;UnDotum&lt;/string&gt;                 &lt;string&gt;은 궁서&lt;/string&gt;&lt;string&gt;UnGungseo&lt;/string&gt;                 &lt;string&gt;은 필기&lt;/string&gt;&lt;string&gt;UnPilgi&lt;/string&gt;                 &lt;string&gt;은 그래픽&lt;/string&gt;&lt;string&gt;UnGraphic&lt;/string&gt;         &lt;/test&gt;          &lt;edit name="antialias" mode="assign" binding="strong"&gt;&lt;bool&gt;true&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="hinting"   mode="assign" binding="strong"&gt;&lt;bool&gt;false&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="autohint"  mode="assign" binding="strong"&gt;&lt;bool&gt;true&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="hintstyle" mode="assign" binding="strong"&gt;&lt;const&gt;hintslight&lt;/const&gt;&lt;/edit&gt;     &lt;/match&gt; &lt;/fontconfig&gt;   How to solve it? Permanent background ssh connection to create reverse tunnel: what is correct way? I see in my company's IT installations that there are numerous disks with one partition occupying all of the space on disk. There is typically something like a /dev/sdb with a partition /dev/sdb1, and the size of /dev/sdb1 is the size of the whole block device. The partition is then mounted with the required file-system format, mountpoint etc.  I do not understand why we would use a partition rather than the whole disk for such (1-partition) configurations. Is there any reason why a partition would be used in this case rather than a whole disk ? Are there any best practices that recommend this approach ?  For example, we could create a Filesystem on a whole raw block device and mount it without needing any partition:  mkfs.ext4 -E stride=16,stripe-width=64 /dev/xvde mount /dev/xvde /mnt/abc   That creates a mountpoint without a specified partition. As a verification:  # sfdisk -l /dev/xvde  Disk /dev/xvde: 6527 cylinders, 255 heads, 63 sectors/track  # fdisk -l /dev/xvde  Disk /dev/xvde: 53.7 GB, 53687091200 bytes, 104857600 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes   For something with a partition, we get the list of partitions under it:  # fdisk -l /dev/sdb  Disk /dev/sdb: 2000.4 GB, 2000398934016 bytes, 3907029168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk label type: dos Disk identifier: 0x00082e8c     Device Boot      Start         End      Blocks   Id  System /dev/sdb1            2048  3907026943  1953512448   83  Linux   My fundamental question is this: Is there some advantage in creating a partition and then creating a filesystem on that partition, as compared to creating a filesystem on the whole disk and then mounting the whole disk ? I use Microsoft Outlook 2013 for work and Google Calendar for everything else. I'd like to find an easy way to sync my work calendar to my personal calendar. Ideally, this would be a software application that sits in the background and automates the syncing.  Requirements   Syncing should be one-way only: Outlook to Google. Must support repeating calendar items. Must allow me to specify which calendar in Google to sync to. I will set up a separate calendar for work items. No intermediate server. Export is direct.   Nice to Haves   Automated. Syncs in real time or regularly (say, every fifteen minutes or so). Do we have negative prime numbers? Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$ Prove by induction: $\sum\limits_{k=1}^{n}sin(kx)=\frac{sin(\frac{n+1}{2}x)sin\frac{nx}{2}}{sin\frac{x}{2}}$ $\sum\limits_{k=1}^{n}sin(kx)=\frac{sin(\frac{n+1}{2}x)sin\frac{nx}{2}}{sin\frac{x}{2}}$  Base case: For $n=1$  $sinx=\frac{sinx\cdot sin\frac{x}{2}}{sin\frac{x}{2}}=sinx$  Induction hypothesis: For $n=m$  $\sum\limits_{k=1}^{m}sin(kx)=\frac{sin(\frac{m+1}{2}x)sin\frac{mx}{2}}{sin\frac{x}{2}}$  Induction step: $n=m+1$  $\sum\limits_{k=1}^{m+1}sin(kx)=\frac{sin(\frac{m+2}{2}x)sin\frac{(m+1)x}{2}}{sin\frac{x}{2}}$  Prove: $\frac{sin(\frac{m+1}{2}x)sin\frac{mx}{2}}{sin\frac{x}{2}}+sin(m+1)x=\frac{sin(\frac{m+2}{2}x)sin\frac{(m+1)x}{2}}{sin\frac{x}{2}}$  Left side: $\frac{sin(\frac{m+1}{2}x)sin\frac{mx}{2}+sin\frac{x}{2}sin(m+1)x} {sin\frac{x}{2}}$  How to prove this equality? I used $sin(u)sin(v)$ identity but that didn't help. How can I do a line break (line continuation)? Dupehammer creates and deletes a comment, causing notifications to go out How to add shortcut keys for java code in eclipse Bash: What does ">|" do? I have just seen this written down;  $ some-command &gt;| /tmp/output.txt   Vertical pipes are used in standard redirects "piping" the output of one command to another, is &gt;| in fact completely useless as it would be the same as just &gt; in this scenario? If $f$ is a periodic function with $T = \pi$, does that mean that $F$ (a primitive of $f$) has the same period ? What about the inverse implication (periodic primitive implies periodic function) ? Why is saltatory conduction in myelinated axons faster than continuous conduction in unmyelinated axons? How does spacing apart sodium and potassium channels allow the action potential to travel faster down the axon? This is the reason always cited for saltatory conduction and myelination, but my mental model of conduction tells me that the density of ion gates along the axon should not affect the speed of the AP.   To illustrate, consider a myelinated axon. A wave of Na$^+$ from action potential site 1, a node of Ranvier, rushes into and quickly diffuses down the axon. (It travels in both directions, but backwards is still in the refractory period.) It diffuses through the myelinated region, its concentration always diminishing. Before it attenuates too much, however, it happens upon node of Ranvier 2, where it triggers another action potential. A new wave of Na$^+$ rushes in and the cycle repeats. This should be plain so far.  Now imagine that there is actually a node of Ranvier halfway between node 1 and 2, called node 1.5. The wave of Na$^+$, on its way to node 2, happens to trigger an action potential at node 1.5, from which a wave of Na$^+$ pours in and either boosts the original wave or replaces it by taking its momentum. Now the reinforced wave proceeds to node 2 and triggers it just as soon as, perhaps even sooner than, if node 1.5 had not existed. Repeatedly insert nodes at higher densities until the situation is simply lack of myelination, and we conclude that unmyelinated axons can transmit an action-potential-triggering wave of Na+ as fast as or faster than a myelinated one.  In short, my point of confusion is this: I cannot see how a higher density of gated channels can possibly slow down the wavefront of Na+ that triggers action potentials. If anything, the additional influxes of Na+ should speed up the all-important wavefront, assuming that new waves really "either boost the original wave or replace it by taking its momentum", and also assuming that the wavefront of Na$^+$ is really all-important for signal transmission, and also assuming that the mere presence of (voltage?) gated ion channels in the membrane does not significantly retard the wavefront.  But the usual explanation for why saltatory conduction is faster than continuous conduction (a fact I hope is empirically and unambiguously established) relies on the putative slowing effect of ion channels on the signal fore. Please explain this effect in more detail, if it is not a misconception. An alternative to translation for non-English speakers? What is the probability that a man likes pink? Empty set does not belong to empty set Herbert in his book "Elements of set theory" on page no 3 says that      we can form the set $ \{ \emptyset \} $ whose only member is $\emptyset $. Note that $ \{ \emptyset \} \neq  \emptyset $, because $ \emptyset \in \{ \emptyset \} $ but $\underline{ \emptyset \notin \emptyset} $·   By the last argument $\emptyset \notin \emptyset$, is he saying that empty set is not a member of or does not belong to empty set  OR  it is a typo and he wanted it to be $ \{\emptyset \} \notin \emptyset $, that set containing empty set is not a member of empty set Why is the year NOT displayed in the questions and comments dates? Why is it that the year is never displayed in the questions or comments? Is it because it is the current year? I, and maybe others, wonder.