I have a r.v. $X$~Poisson($\lambda$). Denote $Y=⌊X/a⌋$. What will be the distribution of $Y$ and how to calculate its mean? (A relaxation can be made if it makes the problem simpler: $a$ is an integer).  What I got so far: $P(Y=k)=P(X∈[ak,ak+a))=Fx(⌊ak+a⌋)-Fx(⌈ak⌉),k=0,1,2,... $, where $Fx$ is the CDF of r.v. $X$.   I can't go any further and derive a closed-form expression for $E[Y]$.  Any help will be much appreciated. Thank you.
If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $\lambda_1, \lambda_2,\ldots, \lambda_n$, what is the distribution of $Y=\left\lfloor\frac{\sum_{i=1}^n X_i}{n}\right\rfloor$ (i.e. the integer floor of the average)?  A sum of Poissons is also Poisson, but I am not confident enough in statistics to determine if it is the same for the case above.
Age and Graduate school I want to write a program for this: In a folder I have n number of files; first read one file and perform some operation then store result in a separate file. Then read 2nd file, perform operation again and save result in new 2nd file. Do the same procedure for n number of files. The program reads all files one by one and stores results of each file separately. Please give examples how I can do it. cmd has wildcard bug? Files in folder:  a.j ab.jp abc.jpg abcd.jpeg   dir command results:  Command       Result (Files shown) ----------    -------------------- dir ?.*       a.j dir ??.*      a.j, ab.jp dir ???.*     a.j, ab.jp, abc.jpg dir ????.*    a.j, ab.jp, abc.jpg, abcd.jpeg   So we can see single ? it means "0 or 1 letter".  Now more dir command results:  Command       Result (Files shown) ----------    -------------------- dir *.?       a.j dir *.??      a.j, ab.jp dir *.???     a.j, ab.jp, abc.jpg, abcd.jpeg --&gt; What is this?! dir *.????    a.j, ab.jp, abc.jpg, abcd.jpeg   In 3rd command why ??? is showing jpeg? Can you explain? Is this bug in cmd? Is it possible to set keyboard shortcuts on a webpage? When I analyze my variables in two separate (univariate) logistic regression models, I get the following:  Predictor 1:    B= 1.049,    SE=.352,    Exp(B)=2.85,    95% CI=(1.43, 5.69),    p=.003    Constant:    B=-0.434,    SE=.217,    Exp(B)=0.65,                            p=.046  Predictor 2:    B= 1.379,    SE=.386,    Exp(B)=3.97,    95% CI=(1.86, 8.47),    p&lt;.001    Constant:    B=-0.447,    SE=.205,    Exp(B)=0.64,                            p=.029   but when I enter them into a single multiple logistic regression model, I get:  Predictor 1:    B= 0.556,    SE=.406,    Exp(B)=1.74,    95% CI=(0.79, 3.86),    p=.171 Predictor 2:    B= 1.094,    SE=.436,    Exp(B)=2.99,    95% CI=(1.27, 7.02),    p=.012    Constant:    B=-0.574,    SE=.227,    Exp(B)=0.56,                            p=.012   Both predictors are dichotomous (categorical). I have checked for multicollinearity.  I am not sure if I have given enough info, but I cannot understand why predictor 1 has gone from being significant to non-significant and why the odds ratios are so different in the multiple regression model. Can anyone provide a basic explanation of what is going on? Do MOSFETs have a diode built into them? I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand. What does the draft mode change? How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$? How can the following inequation be proven?  $$a^2 + b^2 + c^2 \ge ab + bc + ca$$ I am trying to use mathematical induction to prove  $$(2n)!\ge2^n(n!)^2\quad\text{for }n\in\mathbb{N}$$  I am stuck at the $n=k+1$ point. Reputation block layout after earning new privilege On the mobile web after new privilege is earned, reputation block contains some design issues:   no margins poor text color inside bar         By tapping on "track the next one" link, page have been refreshed:   no margins empty block in the middle obscure content         Only after manual refreshing, page looks fine again:        It is hard to reproduce because you need to earn next privilege which takes a lot of time. But I believe it's no too hard for developers :) Can you end Beast Form early? Is there any way to end beast form early? I find myself using it to clear out a particularly hard portion of a dungeon at times, but then I have to sit around for 2-5 minutes so I can be my human character again (I have to feed to keep my health up, so I'm a werewolf longer than the 2.5 minutes).  Is there anyway to end it early? Is doing two PhDs a good path? I am studying a combined bachelor of engineering (electrical) and bachelor of mathematics; I just started this year and will graduate in 2018. The reason why I am doing double degrees and not a single degree is because I love both electrical engineering and mathematics and I could not ignore any of them. So with this in mind, I am thinking of doing two PhDs when I graduate (one in electrical engineering and one in mathematics). Is this a good path or I should concentrate on only one of them? Can I recover my deleted world? I was playing terraria today, and my friend came and deleted the my world.  Is there a way for me to recover my world? Verify proof that sequence converges Image with axis According to this , I thought the new recaptcha API was used network-wide. Though today, when going to  (which for whatever reasons never recognized me automatically), I got this message instead of a usable captcha image.   V1 Unsupported Please direct siteowner to   (Doing a second try doesn't show this message anymore, and the recaptcha v1 works as intended.)  According to , the reCAPTCHA v1 API will stop working on March 31, 2018:  What happens to reCAPTCHA v1? Any calls to the v1 API will not work after March 31, 2018. Starting in November 2017, a percentage of reCAPTCHA v1 traffic will begin to show a notice informing users that the old API will soon be retired. Most websites around the world have already switched to reCAPTCHA v2, where humans pass effortlessly. With our advanced risk analysis engine, reCAPTCHA v2 can effectively separate humans from bots and always stay ahead of the attackers. If your site still uses reCAPTCHA v1, please upgrade to reCAPTCHA v2 immediately. We no longer support reCAPTCHA v1. Any calls to the v1 API will not work after March 31, 2018. Please register a new key and upgrade to v2.  I'm really hoping this will not take  to fix, because we've only got three and a half weeks left! Deductible expenses paid with credit card: In which tax year would they fall? If I paid my business expenses with a credit card during the month of December, which year can I deduct them from my taxes? 2011 (when the purchase was made) or 2012 (when I paid off the credit card)?  IRS pub reference greatly appreciated Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$. I have the following problem:      Let $(X,\tau)$ be a topological space and $(Y,\tau_y)$ be its    subspace, that is $Y\subset X$ and $\tau_y=\{\;A\;\cap\;Y\mid  A\in\tau\;\}$. Let also $F\subset Y$ and $H\subset X$, where $H$ is    closed in $X$. Show that $F=Y\;\cap\;H$ if and only if $F$ is closed   in $Y$.   I solved this problem by assuming that if $F$ is closed then $F=Y\;\cap\;H$, where $H$ is closed in $X$, but I got stuck proving this in the other direction, that is showing that if $F=Y\;\cap\;H$ then $F$ is closed in $Y$.  I tried to do something like this:   Assumption: $F=Y\;\cap\;H$, where $H$ is closed in $X$. If $H$ is closed in $X$ then $X\;\setminus\;H$ is open in $X$ and then $X\;\setminus\;H \in \tau$ which means that $B = (X\;\setminus\;H)\;\cap\;Y\in \tau_y$.  How should I proceed now? Should I investigate the set $B$ now more or what? Word for the behavior of people who complain at everything you do I want to do:  int a = 255;  cout &lt;&lt; a;   and have it show FF in the output, how would I do this?