How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is obvious when we check statistical tables. But how can we show it analytically?  Can anyone share any thoughts?
This question is related to a couple of other questions ( and ) relating to some material that I teach in sampling theory.  Consider the critical points of the  defined as follows.  The value $t_{n-1, \alpha/2}$ is a function of $n$ defined by the implicit equation: $$\frac{\alpha}{2} = \frac{1}{\sqrt{(n-1) \pi}} \cdot \frac{\Gamma(\tfrac{n}{2})}{\Gamma(\tfrac{n-1}{2})} \int \limits_{t_{n-1, \alpha/2}}^\infty \Big( 1+ \frac{r^2}{n-1} \Big)^{-n/2} dr.$$ Normally we treat the input $n&gt;1$ as an integer, but it is possible to extend the treatment to consider it a real value (since the T-distribution is well-defined for non-integer degrees-of-freedom).  I'm trying to come up with a (relatively simple) proof that this critical point value is decreasing in $n$, but I seem to be making quite a mess.  Differentiation using Leibniz rule gets part way, but it quickly becomes a bit of a mess.  Consequently, I would like to solicit proofs of this result --- this simpler the better. Question: How do you prove that this critical point is strictly decreasing in $n$?
Prepositional Phrase - confused - at or in Let's meet in the morning  or at the morning.  in the afternoon or at the afternoon.  in the night or at the night.  in the park or at the park.  working in the background or at the background. Gambler's ruin and coin toss I am looking for a list of all English words that are their own antonyms. Off the top of my head, I can only think of "either", "fast", "to dust" and "to lease", but there must be dozens more. Can you provide a link to a comprehensive list, if such exists?  Also, what are such words most commonly called? I like the term "Janus word" I once heard, but is that widely understood?  Edited by popular demand to make it absolutely clear that this is not a community wiki poll. The editor asked me to use the following format.      Reference   1. Author, blah blah   Instead of      Reference   [1] Author, blah blah ...   Even though in the main text, [1] style has been used.   For example,        Smith [1] has proposed ...      Reference   1. Smith,      This is what I want.   Could you please advise how I can do the above?  Thanks in advance. Why can't I import a SVG file from Adobe Illustrator or Inkscape? Why can't I suggest edits on SE meta sites? When you click &quot;edit&quot; on a meta site as a non-edit-privileged user, the error message appears:  Suggested edits are not allowed on non-tag-wiki posts on meta sites.  Right as I was going to post this as a possible bug, I finally found  which says that per-site Metas don't have suggested edits by-design because it's &quot;not needed&quot;. However, I've been participating a lot over at  where there is a lot still to be figured out in terms of site guidelines.  The Meta is one of the most active I've seen and a lot of the questions are really trying to gather community-built suggestions.  So maybe more of those need to be Community Wiki, but it still brings me to ask why don't per-site-metas &quot;need&quot; suggested edits? Adding bounty: The more betas I participate in, the more I don't get this. Reset password in Drupal 6? Converting a DC motor to a generator Can ANY DC motor become a generator if torque is applied to the shaft? example:   And by any, I mean the really cheap ones too.  Or is there a specific type of motor (of which I do not know the name) that can work in both directions (battery to power the shaft OR torque-on-shaft to create a voltage difference between the terminals). May I use Google Fonts for creating a logo with commercial purposes? I want to create a logo for a contest, the winner will be paid and the winning logo will be the institutional logo of an academic entity, so I need a font which can be used freely or at a cheap cost for commercial purposes.  I see that Google Fonts are delivered as "free fonts" for websites:  May I use them freely for graphic design products (a logo in particular) as well? Reviewer Stats showing wrong count I have totally reviewed 716 suggested edits, but it shows I've approved 486 suggestions and rejected 228.     I searched about it on meta. But, can't get solution. I thought that some of my reviewed posts might get deleted. So, I went to the review history and found      .   It should be a bug. IE must close for event with jQuery to work Title says it all. It seems to be random and is really annoying. Irfanview nearest equivalent for linux Selecting ARIMA orders based on ACF-PACF vs. auto.arima Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint, storm drains clogged with moss, dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint; storm drains clogged with moss; dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights. Walls covered with cracks and peeling paint. Storm drains clogged with moss. Dumpsters overflowing with bags, cans, and rotten trash.  Or maybe the three of them are orthographically correct in their respective contexts? EDIT (based on dockeryZ's answer): I think second example is right, regardless of the context. Explanation: . Where and how does the Bleach anime diverge from the manga? I've seen almost all of the Bleach anime (at least what's been released dubbed), and I'm now interested in reading the manga. However, when the manga closely follows the anime I've already seen, I tend to get bored quickly. Thus, I'd want to pick up the manga at a point right before the anime began to diverge.  At what point in the manga is this? The "Reputation &amp; Moderation" page on any site (/help/site-moderators) still refers to the company behind Stack Exchange as Stack Exchange and not Stack Overflow.     Moderators act as a liaison between the community and Stack Exchange the company.   Seeing as how the name change was recent, I'm not surprised this hasn't been caught yet, and I'm sure there are other instances of this around somewhere. How does the new Tie-Breaker work? In Clash of Clans there was an update on 9/16 adding a Tie-breaker into the wars, can anybody explain in detail how this mechanism works? Proving $\sin(x^2)$ Continuous but Not Uniformly Continuous with non-standard Definition Show that the bounded function $f(x) = \sin(x^2)$ is continuous, but not uniformly continuous, on the interval $[0,\infty)$.  The problem should be solved using the following definitions: Definition of Continuous Function Let $X$ and $Y$ be metric spaces. We sat that a function $f:X\rightarrow Y$ is continuous if given any open set $V \subseteq Y$, its inverse image $f^{-1}(V)$ is an open subset of $X$. Definition of Uniformly Continuous Function Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces. If $E \subseteq X$, then we say that a function $f: X\rightarrow Y$ is uniformly continuous on $E$ if for every $\epsilon &gt; 0$ there exists a $\delta &gt; 0$ such that for all $x,y \in E$ we have $$d_x(x,y) &lt; \delta \longrightarrow d_y(f(x),f(y)) &lt; \epsilon$$ For the continuity part it is clear that $f^{-1}(a,b), (a,b) \subseteq (-1,1) = \{x:f(x) \in (a,b)\}$ which appears to be an open set however I doubt this is sufficient as a proof since it holds for any function $f$. Not sure how to go about proving $f$ not uniformly continuous. Seems to be difficult to prove from the definition alone. Find all natural numbers $n &gt; 1$ and $m &gt; 1$ such that $1!3!5!\cdots(2n - 1)! = m!$   I have been thinking about coming up with some inequalities which would narrow the possible range of pairs $(n, m)$, however the best I have been able to find so far are $m \ge 2n - 1$ and $m \lt n^2$, which is clearly not enough.  Update. The question has tag combinatorics as it is from a book about combinatorics, so there must be at least partly combinatorial solution.