My intuition as to why the statement is true is that the binomial distribution with the higher probability will always have more weight 'further right' if you were to plot it, however I can't turn this notion into a proof.   An equivalent statement is that $$\Pr[A = 0] + \cdots + \Pr[A = k - 1] \leq \Pr[B = 0] + \cdots + \Pr[B = k - 1].$$ All this is saying is that the binomial distribution with the lower probability as more weight 'further left'. Inductive arguments seem to go nowhere...   Any help would be much appreciated.
Let $X\sim B(n,p_h)$ and $Y\sim B(n,p_\ell)$ be two random variables following a respective binomial distribution, where $p_h&gt;p_\ell$. I want to show that  $$P(X\ge\alpha)\ge P(Y\ge\alpha),$$ for any $\alpha\in\{0,1,\dots,n\}$. In other words, I want to show  $$P(X\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge P(Y\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  The statement to be proven is obviously true, but proving it is extremely difficult (at least for me). Can anyone give me some guidance on proving it?  It would be nice if someone can give an algebraic proof too, that is, show that  $$\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge \sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  Thank you very much!
How to run .sh script with keybind Family name pluralization Counting points in polygons using QGIS What is the minimal set of parts for a circut with this AVR microcontroller? My Arduino Uno has a ATMEGA328P-PU microcontroller, and bunch of other stuff on the board. I'd like to program the chip without the Arduino software, and only the minimum of other components.  I want to eventually create things without the cost of Arduino, and I want to learn about the other parts on the board and add them back as as needed.  I'm comfortable with C and gcc, so I can probably figure out the software part. But what, if anything, besides these two products below, do I need to have in the breadboard with the microcontroller? How to view monitor refresh rate in macOS? I just purchased a Dell P2175Q monitor for my 15" MBP w/Touchbar. I'm connecting with a USB-C to DisplayPort cable, but when I go to Apple > System Preference > Displays, there is nothing about the Refresh Rate. Pressing the Option key does not show any additional options.  I've seen countless screenshots where this information should appear. Is there another way to check the refresh rate?  I'm on macOS 10.13.4  Thanks. How to prevent an email from accidentally being forwarded? Is it possible to prevent an email from being forwarded, or to have sensitive information automatically removed when reply-all/forward is pressed?  For example, a Google Calendar invitation includes a link to directly accept/reject the event. However, if someone clicks reply-all, and they keep the HTML formatting, the link will be shared with everyone else who received the invitation, and they can change the user's RSVP. (Specifically, the email is sent only to you, but has a "To:" header showing all invitees, so the problem remains.)  What can be done to ensure that the recipient does not accidentally share the sensitive link? Would it be possible with some MIME trickery or with a semi-standard header? Or is the best practice to refrain from including such "direct action" links when someone might be tempted to reply-all? Multi-Question Questions: How Best to Handle Them? Sometimes I come across a question on SO that is actually a number of questions all in one. If these &quot;sub-questions&quot; are closely related, it's not a problem. But if they are only weakly related (or not at all) it gets ugly. I'm now (ironically or hypocritically) going to ask a few questions of my own for the community to respond to:  Do you consider a question made up of a number of weakly-related sub-questions &quot;inappropriate&quot;? Why or why not? How many is too many sub-questions? What do you think is the best way to deal with such questions?  As an example,  I had some discussion about with another user. I will add a more complete version of my response in an answer below, but in short I felt that this question should be closed and many of the questions asked separately. I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?  \begin{equation} y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial} \end{equation} Programs never open in foreground How can I make programs open in the foreground? Nautilus will open in the background, Remmina (the remote desktop program) also opens in the background. I have to click on the icon in the Launcher for it to come to the foreground. I would think any program that is invoked by any means would come to front, as it opens. It seems if one program invokes another, it will be in the background. If I click on any icons in the Launcher though, it will actually open in the foreground. How do I vertically center a single cell within a table? Bootstrap 3 - Why is row class is wider than its container? How do I create a PDO parameterized query with a LIKE statement? Here's my attempt at it:  $query = $database-&gt;prepare('SELECT * FROM table WHERE column LIKE "?%"');  $query-&gt;execute(array('value'));  while ($results = $query-&gt;fetch())  {     echo $results['column']; } Evaluate the following integral:  $$\int \frac{\sqrt{\sin ^4x+\cos ^4x}}{\sin ^3x \cos x }dx$$ where $x \in \big(0,\frac{\pi}{2} \big)$  Could some give me hint as how to approach this question?  I tried to use the fact that $\sin ^4x+\cos ^4x=1-\frac{sin^22x}{2}$ but it didn't help. How should I proceed? Colorize symbols with lstdefinelanguage in latex I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| = \aleph_i$, then (assuming continuum hypothesis) the cardinality of $2^{\Omega}$ is $|2^{\Omega}| = \aleph_{i+1} &gt; \aleph_i$ and we have $\aleph_{i+1} = 2^{\aleph_i}$.  I am fine with these argument.  What I don't understand is why should there be a smallest $\infty$? Is there a proof (or) an axiom that there exists a smallest $\infty$ and that this is what we address as "countably infinite"? or to rephrase the question "why can't I find a set whose power set gives us $\mathbb{N}$"?  The reason why I am asking this question is in "some sense" $\aleph_i = \log_2(\aleph_{i+1})$. I do not completely understand why this process should stop when $\aleph_i = \aleph_0$.  (Though coming to think about it I can feel that if I take an infinite set with $\log_2 \aleph_0$ elements I can still put it in one-to-one correspondence with the Natural number set. So Is $\log_2 \aleph_0 = \aleph_0$ (or) am I just confused? If $n \rightarrow \infty$, then $2^n \rightarrow \infty$ faster while $\log (n) \rightarrow \infty$ slower and $\log (\log (n)) \rightarrow \infty$ even slower and $\log(\log(\log (n))) \rightarrow \infty$ even "more" slower and so on).  Is there a clean (and relatively elementary) way to explain this to me?  (I am totally fine if you direct me to some paper (or) webpage. I tried googling but could not find an answer to my question) Logistic regression is predicting all 1, and no 0 I am running an analysis on the probability of loan default using logistic regression and random forests.    When I use logistic regression, the prediction is always all '1' (which means good loan).  I have never seen this before, and do not know where to start in terms of trying to sort out the issue.  There are 22 columns with 600K rows.  When I decrease the # of columns I get the same result with logistic regression.   Why could the logistic regression be so wrong?  **Actual from the data**  0 :   41932  1 :   573426  **Logistic regression output**   prediction for 1 when actually 0: 41932 prediction for 1 when actually 1:573426  A**s you can see, it always predicts a 1**   **Random forests does better:**  actual 0, pred 0 : 38800  actual 1, pred 0 : 27  actual 0, pred 1 : 3132 actual 1, pred 1 : 573399 Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction? I oversaw a high-school mathematics test the other day, and one of the problems was the following     Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$   The induction proof is very standard, where the induction step relies on the fact that $\frac{1}{n+1} + \frac{1}{n(n+1)} = \frac{1}{n}$, and I'm sure it's been answered on this site before. However, I got intrigued by the "or other means" part of the question.  I don't know whether the teacher who wrote the test even considered any alternative solutions (he may just have written it so that if anyone has a crazy idea that works out, then they should get a full score for it), but I tried to find one. For instance, we may do the following telescope-ish argument: $$ \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac{1}{n(n+1)} + \frac{1}{n+1}\\ = \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac1n\\ \vdots\\ = \frac{1}{1(1+1)} + \frac12\\ = 1 $$ However, I feel that this is just an induction proof in disguise (hidden in the vertical dots). If one uses the mechanics of the induction proof to check whether the formula is true for a specific $n$, then one certainly does the exact same calculations as I have done here.  Is there a proof of this fact that clearly does not use induction (or at least hides it better)? The more elementary the better, and the ultimate goal would be to do it within the syllabus of the students taking the test (or at least not far from it). For reference, they should be familiar with the summation formula of arithmetic and geometric series and their derivations (so techniques resembling those would be well within specifications).  If there is a solution using calculus, then the students should be able to integrate elementary trigonometric functions, as well as exponential functions, logarithms and rational functions. They are familiar with integration by parts, substitution and partial fractions.  I welcome more advanced solutions as well, of course. MSO Review Queue Privileges are Misleading When I visit Stack Overflow, I see the review option in the bar at the top of the screen.    However when on MSO, I don't see it. I assumed this was because I didn't have enough reputation to access the queue. I was curious what the requirement was on meta, so I moseyed on over to the privileges page. To my surprise, I saw that the privilege is listed as only requiring 500 rep and informs me I already have it:    Confused, I manually typed in meta.stackoverflow.com/review to see what would happen. The page did in fact load, but as you can see meta has no "First Posts" or "Late Answers" queue.    Seeing on that page that a minimum of 2k reputation is required to access any queues on MSO, I report the following bug:  1) On the privileges list on MSO, the 500 rep privilege of "Access first posts and late answers review queues" should not appear.  I also have the following feature request:  2) On the privileges list on MSO, it would be useful if instead there was a 2k privilege that informed you that you have access to the "Suggested Edits" and "Low Quality Posts" queues. The Order of Modification in English Nouns, Preceding or Succeeding? As I don't know the exact linguistic terms, what I mean my "preceding" and "succeeding" in modifying nouns is as follows.  Preceding : delicious food, long way, kind person, et cetera Succeeding : faculty of science, land of despair, a book that changed my mind, et cetera  English seems to accept both ways of modification.  Sometimes, either way will not make a difference. department of science == science department a burning book == a book burning  At other times, changing the order will still make sense but be somewhat awkward or bizarre. Kentucky Fried Chicken ?= Chicken Fried in Kentucky free as in free speech ?= as-in-free-speech free  When writing English, I tend to use more the preceding modification style because my native language Korean is quite strictly precedingly modifying. "빠른 판단을 바탕으로 한 재치있는 응수" directly translates to "a quick-decision-based clever response" rather than "a clever response based on a quick decision"  French as well as other Latin-originated languages seems to be succeedingly modifying. Académie Française, Poulet Frit du Kentucky  while German and Dutch more often use preceding modification.  Well, this mixture in English allows rich style with various ways of expression, but it does often make a non-native English speaker be a little confused. Is there any subtle but general difference between these 2 ways? Or is it just simply case-by-case. Any guidance will help. Local timezones on servers considered harmful? I'm curious as to what other administrators' experiences with timezones, in the context of remotely administered servers, are. In my career I've come across several conventions;   Always, always, always use UTC.   Always, always, always use the timezone of wherever base HQ is. Use the local time of the people who happen to be administering.  Use the local time of the server location.    In some places, I've come across multiple and conflicting conventions. My own preference has been to use UTC, always - with no daylight savings. But for one reason or another, it seems that most people prefer to use some concept of local time, with daylight savings. Although it seems like a straightforward technical matter, discussions around changing conventions always seem to trend towards religious schisms.  What are you using? What do you consider to be the advantages and disadvantages of each approach?