The memoryless property of exponential distribution Let $Z$ be an exponential random variable and $R$ an independent nonnegative random variable. Show that $$ P(Z-R &gt; u | Z &gt; R ) = P(Z &gt; u) $$ From here, I can use Bayes Theorem to split the Left Hand Side, and I know that I have to integrate them.  However, I'm not sure how to exactly to proceed from here. I know that $u \geq 0$.  Any help would be greatly appreciated.  Thanks!
memoryless property of exponential distributions with random variables It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent.
What are the disadvantages of using a Zener diode over a linear voltage regulator? I wonder why a Zener diode is worse than linear voltage regulator.  Some of the posts online mentioned that because of the power dissipating problem. Zener diodes keep consuming a large amount of current while the load may just need a very little amount of current.   If I understand correctly, this phenomenon will happen only when the Zener diode is connected with the load in parallel. What if I connected my Zener diode with my load in series? Also, someone said that linear voltage regulators (e.g. LM7805) are not good voltage regulators as the power dissipated in them is much higher than that in switching voltage regulators. In my understanding, the power dissipated in a linear voltage regulator can be calculated by dropout voltage x current = power converted to heat.   Regarding these 2 points, other than a linear voltage regulator can handler a wider current range, what is the benefit of choosing a linear voltage regulator over a Zener diode? Java Array, Finding Duplicates I'd like to delete a file from my Time Machine backups, but there seems to be no option to do so in the Big Sur version of Time Machine. I go to the Time Machine menu and select &quot;Enter Time Machine&quot;. Then I see the view with the stack of windows. I browse to a file and select it. Now I want to delete it so it doesn't exist and take up space in backups. There is no option to delete it - either the action menu in the toolbar, or the control-click context menu. How do I delete a file from Time Machine backups? There used to be a way to do this. (I'd take a screenshot, but the key shortcuts for screenshots don't work in Time Machine. Not sure if this is just a Big Sur thing or if that's always the case.) Update: I should mention I'm using APFS for my Time Machine volume. That seems to be relevant. Also, I contacted Apple support about deleting entire snapshots (slightly different goal than the one here in this question). They said you can't do it anymore. It seems like there's some bugs here, because I have a &quot;Move to Trash&quot; option if I control-click one of my backups. It seems to work initially but then the backup comes back.  I don't like losing this much control over my files. I may look for another backup solution. Lots of people would like to visit the EEA area to take holiday or to visit family or even to go for longer term visa arrangements in the Schengen zone.  In order to do this, they need a Schengen (or national) visa.  This appears to be a fairly straightforward process, but hints and tips can always be helpful.  Without focusing on any individual's circumstances, what are some helpful hints that can improve the likelihood of a successful application? Why are the Degrees of Freedom for multiple regression n - k - 1? For linear regression, why is it n - 2? I'm reading a textbook and I see this question:    So there are 200 women, and the DF is 196, implying that the equation for DF is $n - k - 1$. There are 3 variables: bp, age, and type so $k == 3$. What's the intuition behind this?  Also, why is the degrees of freedom for linear regression n - 2? What is the range of the validity of Coulomb's law? How to do selective-color in Photoshop? How to make the single color in a photography to be color and other parts changed to B&amp;W?.  Don't know exactly the name of the effect, to be more clear in the following photo I want the green part to be green and other parts( leg and sandal) in B&amp;W..    How to achieve that effect in Photoshop? For any matrix $A$, $AA^T$ is symmetric, and thereby diagonalizable (Spectral Theorem). This implies that $AA^T$ has n linearly independent eigenvectors and therefore full rank.  Does this imply that for any matrix $A$, $AA^T$ is always invertible ? I know there is something wrong here because from the above chain we can conclude that a symmetric matrix is always invertible, which is definitely not true.  Could someone please point out where I am going wrong? I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you. I want to programmatically create a site using custom web templates in sharepoint 2010.  In 2007 I used to get a   SPWebTemplate template = site.GetCustomWebTemplates(LCID)[number];   and used this SPWebTemplate in  SPWeb newSite = web.Webs.Add(url, title, description, LCID, template, true, false);   This worked quite well for 2007. But in 2010 the site templates are saved into the solution store  SPSite.Solutions   which only returns SPUserSolution objects and not SPWebTemplates.  So my question is: How can I programmatically create sites using the site templates from the solution store.    UPDATE: I tried to use  SPWeb newSite = web.Webs.Add(url, title, desc, LCID, templateName, true, false);   where templateName is a string like '{Template-GUID}#TemplateName', but it still does not work. It throws the following error:  Critical error: File or arguments not valid for site template '{4a2d8952-a0af-49a6-8f55-46a59c8d68a2}#CustomerSiteTemplate'. Parameter name: WebTemplate    Any other ideas how to create a site based on a SPWebTemplate? Long-distance Mind Trick In the Star Wars universe, could a Jedi Mind Trick be performed over the telephone?   For instance, could Yoda have called up the Imperial embassy on Dagobah and mind tricked whoever answered into getting him in touch with Palpatine? (I know there's no embassy or telephones on Dagobah, but serious question nonetheless.)   To clarify, are long distance mind tricks not only possible but would being in communication with another person, through telecommunication, allow a force user to better perform the action? sum of the residue at all poles of the rational function p/q is 0. We are given two polynomials p, q with deg(q)>1+deg(p). Show that the sum of the residue at all poles of the rational function p/q is 0.   How do I go about solving this? Sphere rolling up a ramp In this question, I did energy conservation, with the states being the initial one and the one where it is at the top , and I got $$ .5 I w^2 + .5 mv^2 = mgh$$ However, in  the answer key they have done, $$ .5 mv^2 + .5 Iw^2 = .5 Iw^2 + mgh$$ Now why would the k.e be turned to p.e but not the rolling energy?? Short story identification: society partitioned off by high wall Reading another question today reminded me of a short story I read in the early 90s (probably written in 60s or 70s though). A society is blocked off by a really high wall; the story follows a girl who tries to get over the wall. I think she manages it with a hot air balloon. When she reaches the other side she meets an older man who had managed to scale the wall in his youth - his solution had been to use a flock of birds to carry him.  UPDATE  I seem to recall that it splits off an elite group of people - getting over the wall is like a rite of passage. Most people don't even bother to get over it (or even be concerned by its existence at all), but a few try. There might be some magic involved (on the other / better side), or they might just be more advanced technologically.  FINAL UPDATE  As pointed out in the answer, and confirmed by me re-reading the story, I was mistaken in a few details. The main protagonist is not a girl, it's a boy called Porgie. He doesn't use a balloon, it's a glider he made, assisted with a broom. Magic is involved, but it's on the inside, with technology on the outside. Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism. Strange behaviour of 'sudo chmod -rwx': Does not remove group write permissions I am working to integrate unit testing into the development process on the team I work on and there are some sceptics. What are some good ways to convince the sceptical developers on the team of the value of Unit Testing? In my specific case we would be adding Unit Tests as we add functionality or fixed bugs. Unfortunately our code base does not lend itself to easy testing. What is the difference between the following class methods?  Is it that one is static and the other is not?  class Test(object):   def method_one(self):     print "Called method_one"    def method_two():     print "Called method_two"  a_test = Test() a_test.method_one() a_test.method_two() Looking for a short story about a robot evolving to have supernatural powers I'm looking for a short story from about 50 years ago about a robot who escapes after being slated to be melted down and discovers while in hiding that it has evolved to have some supernatural powers. overbrace/underbrace but with an arrow instead Is there a similar command to \overbrace{foo}^{bar} but instead of braces, an arrow pointing up/down?  EDIT: Here's a little demonstration of how I'd expect something like this to work:    And possible syntax would be \overarrow{a=b}^{\text{theorem X}}