Given a continuous random variable $X$, what are the units of the PDF and CDF of $X$?
On the , there is this line:      $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)    How can a value $&gt;1$ be OK? I thought all probability values were expressed in the range $0 \leq p \leq 1$. Furthermore, given that it is possible to have such a value, how is that value obtained in the example shown on the page?
How do I force Python's print function to output to the screen? When I'm on my mod site I have my diamond dropdown:    But on Meta.SE and other sites, I have the StackBurger:    This doesn't show on my mod site, so I can't switch between sites easily. Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$ Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$. It is the distance of a point in a triangle from its vertexes which gives the minimum when $(x,y)$ is on the Fermat's point.  But now I am stuck in finding the location of Fermat's point.  $(1) \quad 4$ $(2) \quad 5$ $(3) \quad \sqrt{3}+2\sqrt{5}$ $(4) \quad 4\sqrt{2}$ How to show every field is a Euclidean Domain. I'm having trouble proving this. This is what I have so far:  Let $F$ be a field. Let $v(x) \rightarrow 1$ for all $x$ not equal to $0$.  So if we let $x$ be in $F$ where $x$ not zero then we can write $x$ as: $x=qy+r$ for some $y$ in $F$. If $r$ not zero then $V(r)=1$.  Not sure what to do from here. I know eventually I'm supposed to get no remainder $(r=0)$ but I'm stuck at applying the definition and valuation map. What exactly goes into a transformer's VA rating? At a fundamental level it seems that there has to be a limit to what the VA rating can represent.  For example if I draw 2 Amps-RMS from the secondary of a 30VA rated transformer at 15 Vrms this makes sense as the maximum Complex Power I can draw. Still, at a power factor of 1 my maximum current in the transformer would be sqrt(2) * 2 Amps.   But what if my power factor is abysmally low (ie in a DC rectifier) due to harmonic distortion and while I still draw 2A-RMS I have brief current spikes of 15 Amps? Surely the transformer's magnetics couldn't handle these spikes as I'd expect the core would saturate.  My question is what is the practical limit for VA ratings of transformers? Would brief current spikes that reduce to only 2A-RMS in this case be fine? Obviously the secondary/primary copper losses play a part in the rating but how do the magnetics factor in?  I ask this because I am designing a simple bench power supply using a 125VA transformer with a 24Vrms secondary and am trying to figure out how much power I can draw before the current spikes into the inductor are too much for the transformer. I also don't want to have to resort to active PFC as I don't want to risk messing up any HVAC PCB design.  Thank you Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values Solving a Diophantine equation with a large solution Placing nodes midway on paths using |- or -| in tikz I make frequent use of |- and -| with computed points in diagrams, like so:  \coordinate (a) at (0,0); \foreach \x in (1,...,10) {     \coordinate (b) at ({whateverFunction(\x)},{whateverOtherFunction(\x)});     \draw (a) |- (b); %Straight line segment     \draw (b) -| (a); %Another straight line segment     \coordinate (a) at (b); }   To get plots like:   I would like to be able to label the midpoint of each line segment, but doing something like node[midway,above] {a} does not act like expected:    I want something like:    Why does midway not work, and what is a good workaround here? I'm studying measure theory and two question came to my mind:        If $f:\mathbb{R}\to\mathbb{R}$ is monotone and $B\subseteq\mathbb{R}$ is borel, is the image $f(B)$ borel?   If $f:\mathbb{R}\to\mathbb{R}$ is a monotone function (say, non-decreasing), does there exist a sequence of continuous functions $f_n:\mathbb{R}\to\mathbb{R}$ converging pointwise to $f$?      Here's the motivation for those questions: Let $(X,M)$ and $(Y,N)$ be measure spaces.   It's known that if $\mu$ is a measure on $(X,M)$ and $f:X\to Y$ is measure, then we have the pushforward measure $f_*\mu(A)=\mu(f^{-1}(A))$ on $(Y,N)$. What if we were to define a "pullback measure"? Given a function $f:X\to Y$ such that $f(M)\subseteq N$ (i.e. $f$ maps measurable sets to measurable sets) and a measure $\nu$ on $N$, the natural formula would be $f^*\nu(A)=\nu(f(A))$. So we ask if is there a good amount of functions which map measurable sets to measurable sets in $\mathbb{R}$, and monotone functions seem like good candidates (for strictly monotone, continuous functions, the result is valid and there are several answers on the web). If this were true, maybe we could use some convergence argument to solve the problem above. How many ways can the average of n dice be a? How can I make a hexagonal grill? How would I be able to create a grill of sorts found on the front of some computer cases such as this one lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. Is it possible, on Debian with Klipper running, to have a separate "clipboard stack" for a particular application?  I'd like the text that I copy in application Y not to be shared with all the other applications running.  All the solutions mentioned at  seems like overkill. Looking for a quick and dirty way to automate . I presume this could be done with an automation to press the mouse button from two separate pixel coordinates, with delay, so long as the browser window doesn't move -- unless there's a better way.  I tried building a Watch Me Do workflow, in Automator, but can't seem to get it to work.  I get the error "application Chrome is not running when it is expected to." It also never lists mouse clicks for some reason.  I do have accessibility set up, in system preferences > security &amp; privacy > privacy > accessibility.  Automator, Chrome and System Events are all checked within there (among other stuff of course).  I tried the markashworth workaround as detailed  -- which also doesn't seem to work. Using his method, I get this error:    UPDATE: I just downloaded "cliclick" and figured out how do both clicks as a terminal command, with the proper delays.  BUT -- I don't know code and need a way for this terminal command to repeat "x" amount of times.  Anyone know how I could do that in Script Editor?  The command line is:  cliclick c:132,286 w:100 c:132,286 w:5000 c:1100,238   I would need a few seconds of delay before the loop repeats, which I presume I could do with a final "w:" in that line.  Thanks for the help!! How to add item to right-click menu when NOT selecting a folder or file? How do I add a menu item to my context menu to open up powershell (or any other arbitary application) when I right-click inside Windows Explorer?  I found tons of links on how to add menu items when doing this: (selecting a folder; )    ...but couldn't find any instructions on how to add menu items when doing this: (not selecting a folder or file)    I did find  and tried adding powershell to HKEY_CLASSES_ROOT\Directory\Background\shell\, (loosely based on the  and linked above) but it didn't work for me.  I'm running Windows 7, by the way.  Also, I'm currently interested in adding only PowerShell, but I'd prefer a general solution I can apply to add any program to the context menu. Create a .desktop file that opens and execute a command in a terminal Evaluate $6^{433} \pmod {21}$ and a proving question Question 1: Denote $a \mod b$ as $a \% b$, where $a$ and $b$ are some integers  Evaluate 12^32475 % 21    The following is what I tried:    12^32475 % 21 = (12^3 % 21)^10825 % 21 = 6^10825 % 21 = ... = 6^433 % 21   But how to continue ?  Question 2:  I don't understand this: n-1= -1 mod n. Where is this from? And does this line become the next line?    Context: my class is discrete math for computer science. So far for related topics, it covers modular inverse, (extended) GCD algorithm, Fermat's Little Theorem and Chinese Remainder Theorem (CRT). How to customize the package editor interface? How can we discipline a strong-willed, misbehaving 5 year-old? My 5 year old son receives and fears punishment but repeatedly exhibits the same unacceptable behaviors.  He's always been strong-willed, and is usually the kind to view a statement like "Don't do that" as an open invitation to do it.  We've tried redirection or distraction tactics, but he is still so narrowly focused on doing the original behavior.  Even worse, he will typically do the prohibited behavior and flagrantly tell us or show us that he's done it.  Repeatedly, even after punishment.  (He's in time out right now for this very thing).  He's consistently punished, yet he continues down the same self-destructive path each time.  When he is punished, we explain what he did wrong, why he's being punished, and why he shouldn't do what he was told not to do.  Sometimes he'll even tell us first why he's in trouble, so it makes me wonder if he's acting out for the inevitable negative attention he gets.  I think he gets plenty of love and positive attention- he's an only child and stays with a relative during the day.  We're trying to give him a good, fun childhood, but we need to figure out how to effectively discipline him.    His repeated misbehavior includes:   Yelling at the top of his lungs in public (stores, friends' houses, etc) Running away from us (at parks, stores, parking lots, etc) Talking back and disrespecting authority figures Hurting animals even after we explain why it's wrong Repetitive nuisance behaviors (repeating himself, banging his fists, kicking walls, intentionally crumbling food on floors, etc) Repeatedly asking us why he can't do something even after it's been explained to him Basically, most bad behaviors that young children may do, but repeated over and over again, even after discipline   He gets a warning to stop the behavior, then depending on the severity of what he's done (and where we are), he'll either go to time out or lose a privilege (such as taking away a favorite movie or missing out on an activity).  Going to time outs and losing privileges really upset him, but still do not ultimately deter him from the bad behavior.  It's like he can't help himself.  My spouse and I are getting worn down.  We try to model good behavior and encourage him to be well-mannered and respectful.  This behavior mortifies us.  It seems we can't even enjoy life as a family because our son is so disruptive and unresponsive.  Most public outings have to be cut short or eliminated because he consistently misbehaves and doesn't respond to our discipline.  After he's misbehaved, when we try to have heart-to-heart talks with him about his behavior and why we expect him to mind us, he usually has a little trouble expressing himself, and a lot of times we get the same nonsensical responses from him (Us: Why did you do [bad behavior]?  Him: Because I didn't want to).  He also doesn't like to make eye contact.  We don't usually feel like we've gotten through to him.  Is there another discipline tactic we could try?  Or does it sound like we need to see a professional at this point?  We're at a loss, and don't want things to spiral further out of control as he gets older (and even begins school).  We just want him to understand that his behavior isn't always acceptable.  Thanks for reading.  Update (from comments below): My spouse and I disagree on spanking, as we had different experiences with it. He feels it's necessary sometimes. Since our son failed to improve after time outs and taking away items/privileges, I grew to agree with him and we used it for select bad behaviors. It didn't work any better. However, when he was spanked, he was always told why. I didn't mention that we have used it because I was afraid it would distract from the original question, but since it has already come up [in comments], yes, it has been explored, but with no better results than our other tactics. Groups of order $pq$ without using Sylow theorems