Suppose that $U,V$ are two independent uniform $(0,1)$ random variables. Find the distribution  of $X=\dfrac{\max\{U,V\}}{\min\{U,V\}}$.  Ok to find $\max(U,V)$ we have $P(\max(U,V)\leq r)=P(U\leq r)P(V \leq r)= r^2$ so $P(\max(U,V)=r)=2r$. I can do the same trick with $\min(u,v)$ but then I don't know how to deal with the fraction of the two?
Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution function of $Z$. Can anyone give me some hints on how to proceed? I can only note that $\mathbb{P}[Z\geq 1]=1$ and $$F_Z(t)= \mathbb{P}[Z \leq t]=\mathbb{P}[X\leq Y, Y\leq tX]+\mathbb{P}[Y \leq X, x\leq tY]$$
I have a 2 year old son and another baby on the way. I just recently found out I'm pregnant but not sure how far just yet. My son rammed into my stomach like he is always doing but this time it made me start having small pains in my stomach. I rushed to the bathroom to make sure there wasn't any bleeding.  I'm concerned his rough play could hurt the fetus if it continues. How should I put a stop to it? Lubuntu Installer Giving Error after Partition creation "YOUR SYSTEM MAY OR MAY NOT BOOT " I was replacing my Ubuntu 18.04 with Lubuntu 20.04. I wanted to keep my /home partition same. After creating /root , swap and changing mount point for /home partition the installer gave a warning that /boot/efi partition is not created Lubuntu 20.04 needs /boot/efi partition to boot. Please create a FAT32 partition and mount it as /boot/efi and enable flag esp. Then I created the /boot/efi  partition but there was no esp flag(s). I had installed Ubuntu 18.04 alongside windows10. What should I do now? I'm currently working on some logging code that supposed to - among other things - print information about the calling function. This should be relatively easy, standard C++ has a type_info class. This contains the name of the typeid'd class/function/etc. but it's mangled. It's not very useful. I.e. typeid(std::vector&lt;int&gt;).name() returns St6vectorIiSaIiEE.  Is there a way to produce something useful from this? Like std::vector&lt;int&gt; for the above example. If it only works for non-template classes, that's fine too.  The solution should work for gcc, but it would be better if I could port it. It's for logging so it's not so important that it can't be turned off, but it should be helpful for debugging. Prove or give a counterexample:  $$(A ∪ B = A ∪ C) \land (A ∩ B = A ∩ C) \implies B = C.$$  I think this is true, but I am not sure how to show it. I don't know if there are any manipulations with unions and intersections that allows me to move things around. Thanks in advance for any hints. If I roll two fair dice, the probability that I would get at least one 6 would be.... Alright, I'm toying around with converting a PIL image object back and forth to a numpy array so I can do some faster pixel by pixel transformations than PIL's PixelAccess object would allow.  I've figured out how to place the pixel information in a useful 3D numpy array by way of:  pic = Image.open("foo.jpg") pix = numpy.array(pic.getdata()).reshape(pic.size[0], pic.size[1], 3)   But I can't seem to figure out how to load it back into the PIL object after I've done all my awesome transforms.  I'm aware of the putdata() method, but can't quite seem to get it to behave. How to tell which row number is clicked in a table? Advice/literature on combining items with different response scales into composite scales? Let's say I have some self-report items measured on a 5-point Likert scale (Strongly Disagree to Strongly Agree) and other items measured on a 4-point Likert scale (Never, Rarely, Sometimes, Often). Can anyone point me to literature (or practical advice) on combining these items into a composite scale? Let's assume for the sake of argument that we have some empirical evidence that items should be combined.  Some ideas:  1. Sum raw scores  Pro: Easy  Con: Max responses on the 5-point scale (originally 4's) drive up the total scale score more than max responses on the 4-point scale (originally 3's).  2. Rescale and sum  Put all of the items on a 0-1 scale and sum. So the 4-point items (0,1,2,3) would be multiplied by (4/3)/4 and the 5-point items (0,1,2,3,4) would be multiplied by 1/4, resulting in possible values of (0,.33,.66,1) and (0,.25,.50,.75,1), respectively. This way max responses on the 5-point scale (originally 4's) would not drive up the total scale score more than max responses on the 4-point scale (originally 3's).   Pro: Items would have equal weighting. (could be a con, depending on your perspective).  Con: Ignores differences in variability between items on different metrics?   3. Standardize and sum  A related approach would be to standardize all of the items (z score) and then sum.  Pro: Addresses differences in variability between items on different metrics  Con: Total scale score becomes less interpretable and sample-specific. The latter makes it hard to benchmark as a measure to be used in other settings/other samples.  4. PCA or other data reduction   4a. EFA to get factor loadings. Multiply scaled items by factor loadings.  4b. PCA to get score of first principal component.  Pro: Items weighted by influence.  Con: Same as #2. EFA-derived scores could vary a lot depending on rotation/extraction choices. Some would not advise on ordinal data.    Overall: I like #2 because it seems easier to compare results across different samples. Thoughts? Alternative ideas or concerns about the ideas presented? Trouble trying to use Windows Search to search for file with a question mark in the content It seems that Windows Search is having trouble when I specify a question mark character as the content that I'm searching for within my files.  I've tried:   content:? content:"?" content:"\?" content:\? content:%3f content:"%3f" content:\3f content:"\3f" content:\00\00\00\3F   but each of these yield no results when I do indeed have files that contain ? within the scope of the search. If I instead search for something like content:happy it will indeed find all files with the phrase "happy" within those same files.  My assumption is that Windows Search is treating some characters such as the question mark as a special character likely for the purpose of wild card expressions. To test this theory I've also searched for files containing an asterisk * and indeed I have the same issue as when searching for content of question mark. What I'm wondering is whether there is any way to get a search to work where I can search for a literal expression without any type of wildcard matching or at least a way to escape the special characters within the expression.  By the way I'm using Windows 7 Enterprise. If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$  I tried using the fact that  $\frac{\cos A}{1-\sin A}=\tan(\frac{\pi}{4}+\frac{A}{2})$ but now not able to eliminate $\cos \alpha$ and $\cos \theta$. How should I proceed? Why are text format commands used inside braces? Most commands in Latex are written with the arguments inside braces following the command: \command{arguments}. But text formatting options, like \em, or \scriptsize are usually written inside the braces, like {\em some italicised text}. Why is this? What makes these commands different? I'm particularly asking because it seems to make standardisation - of which LaTeX really appears badly in need - more difficult. How can I find out whether my install of Windows (specifically XP and Vista) is either 32 or 64 bit?  Additionally, how can I find out if my hardware will support a 64-bit operating system? Is there a "sentence stretch" that acts horizontally in a similar way as \setstretch acts vertically? How can I determine the name of the Bash script file inside the script itself?  Like if my script is in file runme.sh, then how would I make it to display "You are running runme.sh" message without hardcoding that? Resetting Ubuntu to factory settings why is the output of the following code coming to be zero? Stop Mathematica from giving imaginary solutions Not getting edit's +2 rep after reaching 2k rep After reaching 2k rep , I'm not receiving  +2 rep per edit. Why? Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.  Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$  I get to:  $$f_{n+1}^2 + f_{n+2}^2$$  Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?  Thanks! Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$ $${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$$  I see I should use Riemann sum, and that $$\int_0^{\infty} {dx \over {1+x^2}} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}}$$  But how do I explain this exactly, and how to get the other half of the inequality (with ${\pi \over 2} + 1$)?