Why was Luca Brasi's life essentially wasted in The Godfather?
Vito and Luca question
A random variable with neither probability density function nor probability mass function... is this example wrong? This is from Shreve's Stochastic Calculus for Finance II, Appendix A.3. Consider infinitely many independent coin tosses with outcomes $\omega_n$, $n = 1, 2, \dots$ such that $\omega_n$ can be either $H$ = heads or $T$ = tails. Define \begin{equation} Y_n\left(\omega\right) = \begin{cases}1 &amp; \text{ if }\omega_n = H \\ 0&amp; \text{ if } \omega_n = T\text{.} \end{cases} \end{equation}  Define $\displaystyle Y = \sum\limits_{n=1}^{\infty}\dfrac{2Y_n}{3^{n}}$. Shreve shows that $Y$ can only take values in the Cantor set, and since that particular set has measure zero, its Lebesgue measure is zero as well, thus it does not have a probability density function (since integrating such a density function would give a value of $0$). This part I understand, as well as my professor.  Now Shreve claims that $Y$ does not have a probability mass function.      If it did, then for some number $x \in C$ [the Cantor set] we would have $\mathbb{P}\left(Y=x\right) &gt; 0$. But $x$ has a unique base-three expansion    \begin{equation} x = \sum\limits_{n=1}^{\infty}\dfrac{x_n}{3^{n}}\text{,} \end{equation}   where each $x_n$ is either 0, 1, or 2 unless $x$ is of the form $\dfrac{k}{3^{n}}$ for some positive integers $k$ and $n$. In the latter case, $x$ has two base-three expansions... In either case, there at most two choices of $w \in \Omega_{\infty}$ [that is, the infinite coin-toss space] for which $Y(\omega) = x$. In other words, the set $\{\omega \in \Omega; Y(\omega) = x\}$ has either one or two elements. The probability of a set with one element is zero, and the probability of a set with two elements is $0 + 0 = 0$. Hence $\mathbb{P}\left\{Y = x\right\} = 0$.   My professor and I were able to follow this until the statement "The probability of a set with one element is zero." Is this a true statement? How so?   I thought it would be perhaps the fact that the Lebesgue measure of $[a, a] = \{a\}$ is zero, but I don't see any reason why Shreve would only use the Lebesgue measure, since it is not a general probability measure. How to test @future methods with setup objects and avoid MIXED_DML_OPERATION Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.  I think i should use Liouville's theorem, but how can i do that? Help! How to plot the first n rows of a table using pgfplots? What's the scope of a variable initialized in an if statement? What open source software for auto-alignment of photographs? Do you know any open source tool to automatically align images, similar to the auto align feature in Photoshop? How to roll back Ubuntu to a previous version? I just installed a new version of Ubuntu and I want to roll it back to the previous version.  How can I do that? Is that even possible? How do I check if a list is empty? How to create a batch job, numbering images (writing text on them) from 1 to 100 in Photoshop? Looking to create a 100 images that would have their appropriate numbers written on them.  Since I`ll be most likely making changes to the template in the future I figured it would be better to automate it somehow.  Is there a simple solution to this problem? Why would Mystique's DNA allow for Sentinels that can duplicate mutant powers? The future Sentinels in  are clearly shown adapting to powers thrown at them by copying them. Isn't that what Rogue's DNA would allow for, not Mystique's? Mystique can just change appearance, not copy powers ... right? So why would study of her mutant genes allow for the abilities of the future Sentinels? Is there a way to convert a List(of Object) to a List(of String) in c# or vb.net without iterating through all the items? (Behind the scenes iteration is fine – I just want concise code)  Update: The best way is probably just to do a new select  myList.Select(function(i) i.ToString()).ToList();   or  myList.Select(i =&gt; i.ToString()).ToList(); Show (within ZF Set Theory without choice) that for any sets $A,B,C:$   If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$   That is, if there is a bijection from $|A|^{|B|}$ to $|B|$, then  $|A| = 1 = |B|.$  Please help me.. What does the register keyword do in C language?  I have read that it is used for optimizing but is not clearly defined in any standard.  Is it still relevant and if so, when would you use it? Is Responsive Design good for web based software? Centering polygon on specific coordinates using QGIS I need to place a set of polygons onto a set of points, so that each polygon is centered on its corresponding point. How do I do that? In MapInfo I can just manually edit the coordinates of the center, but there does not seem to be an equivalent in QGIS. I'd like some suggestions for good reading materials on the Keldysh formalism in the condensed matter physics community. I'm familiar with the imaginary time, coherent state, and path integral formalisms, but lately, I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism at least in that one can evaluate non-equilibrium expectations. Getting absolute path of a file in my earlier question you can find at:   i describe my error:     I'm using an Asus Zenbook UX562F with an integrated GTX 1050 Max-Q.   After the installation, I installed Nvidia-390 and updated the kernel   with ukuu to 4.20.17 to fix no sound issue.      After the reboot, everything went well and I did:      apt-get update apt-get upgrade + J      All was done well, so I restarted and on the next boot after selecting   Ubuntu in GRUB it gave me a long black screen (violet screen) where   nothing changes after so much time.      I tried to add nomodeset and noveau.modeset=0 single and together but   didn't help so I don't know what to do.   After investigating deeper by installing packages and try to boot i found out that the package which is causing the error is:  intel-microcode   when i purge and block this package with:  sudo apt-get purge intel-microcode sudo apt-mark hold intel-microcode   before upgrading wuth  sudo apt-get upgrade   my system is able to boot normal.  The problem i see is that as i understand it the package protects my intel cpu  Intel® Core™ i7 8565U 1.8 GHz   is vulnearable to meltdown attacks which get fixed by the package.  What can i do to fix the vulnearbility and also get my system to a bootable state?  Thanks for your help Vimu  Edit: I used the manual update technique mentioned at (in german):    and i used the files from the intel github repo:    with the following code mentioned in the tutorial to update the firmware microcode files  sudo cp intel-ucode/06-8e-09 /lib/firmware/intel-ucode/ echo 1 | sudo tee /sys/devices/system/cpu/microcode/reload   i overwrote the old microcode firmware with the newest available on the intel repo  it leada to a bootable system  will this be the best possible fix without upgrading the intel-microcode package provided by apt? Let $f$ be a continuous function. If $f$ is uniformly continuous then it maps cauhy sequence into cauchy sequence. Is the converse true? That is if $f$ maps cauchy into cauchy gives the uniform continuity of $f$. I think it's no. Can suggest a counter example Salesforce1 on focus scroll to the top