Hey guys can you help me with this? The problem setting is that X and Y are independent random variables and Z = Y / X. I want to verify that: $\displaystyle f_ Z(z)\, = \int _0^\infty {{y}} f_ Y(y)f_ X(yz) \, dy$ In other words, is this true? I hope the problem is clear. Thank you in advance. PS I know the answer but I'm looking to understand.
Suppose $X$ and $Y$ are independent positive random variables with probability density functions $f_X$ and $f_Y$ respectively. Show that $Z=X/Y$ is absolutely continuous and find its probability density function.   $\textbf{My Thought:}$ In order to show that $Z$ is absolutely continuous, I need to show that it has a commulative distribution function. Then I can also differentiate the latter to obtain the probability density function of $Z$. I have the following calculation \begin{align*} P(X/Y\leq z)&amp;=P(X\ge zY,Y&lt;0)+P(X\le zY,Y&gt;0)\\ &amp;=\int_{-\infty}^{0}\left(\int_{yz}^{\infty}f_{X}(x)dx\right)f_{Y}(y)dy+\int_{0}^{\infty}\left(\int_{-\infty}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy\\ &amp;=\int_{0}^{\infty}\left(\int_{0}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy. \end{align*} Also, if $z&lt;0$, then we have that $F_Z(z)=0,$ since $X,Y$ are positive random variables. Therefore the random variable $Z$ is continuous, and we can differentiate it to obtain the probability distribution function of $Z$. We have that $f_Z(z)=0$ if $z\leq 0,$ and otherwise we have $$f_Z(z)=\int_{0}^{\infty}yf_{X}(yz)f_Y(y)dy.$$    Is my reasoning above correct?  Any feedback is much appreciated.  Thank you for your time.
Particularly, does proc always list the VmRSS value in kB? I can't find a solid answer on the , although it appears that it sticks to one unit. Proof that in a field $x=0$ is equivalent to $x=-x$ Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$. But how do we know that $1\neq -1$ for an arbitary field? I want to make a logistic model from my survey data. It is a small survey of four residential colonies in which only 154 respondents were interviewed. My dependent variable is "satisfactory transition to work". I found that, of the 154 respondents, 73 said that they have satisfactorily transitioned to work, while the rest did not. So the dependent variable is binary in nature and I decided to use logistic regression. I have seven independent variables (three continuous and four nominal). One guideline suggest that there should be 10 cases for each predictor / independent variable (Agresti, 2007). Based on this guideline I feel that it is OK to run logistic regression.    Am I right? If not please let me know how to decide the number of independent variables? How to shift EXIF date/time created by time in days, hours, minutes? I have two camera bodies, I was shooting while on vacation, when I downloaded the photos to lightroom, one body was set to the correct date/time, the other was off by 722 days, 4 hours, 32 minutes.  Is there a tool available that can help me fix the date/time so that they appear in sequence while browsing my lightroom library?  I'd like for them to be as close to the right time.     I was able to adjust created year, month, date, but it doesn't appear lightroom will allow me to adjust increments (only hour increments).  Many thanks if you can help me in this task! Taylor Expansion $\log(\frac{1+z}{1-z})$ If I have an item, combined with another item, is it still a singular subject in the sentence? For example:        This paper, combined with some glue, makes for an afternoon of art projects.   This paper, combined with some glue, make for an afternoon of art projects.      All of my experience tells me that it should be the first, but I have multiple people arguing for the second. Cohen's d for 2x2 interaction I have a significant 2x2 interaction in a between-subjects design.  I would normally report omega-squared as my effect-size measure, but I've been asked to provide Cohen's d instead?  Can Cohen's d be computed for an interaction?  If so, how? SEO preference: does a site with WWW rank better than without WWW? What is the distribution of primes modulo $n$? If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant Let $\Bbb D$ be the open unit disc and $f:\Bbb D\to\Bbb C$ be an analytic function such that $|f(z)|\le |f(z^2)|$, for all $z\in\Bbb D$. Prove that $f$ is constant.   Here is my proof:   For any $0&lt;r&lt;1$, consider the closed disc $B=\{z\in \Bbb D:|z|\le r\}$. Then the given condition implies that $|f(z)|\le |f(z^{2^n})|$, for all $n\in\Bbb N$ and for $z\in B$. Since for $z\in B$, $|z^{2^n}|\le r^{2^n}\to 0\implies z^{2^n}\to 0$. So by continuity, $|f(z)|\le |f(0)|$, for all $z\in B$. So by the Maximum modulus theorem, $f$ is constant on $B$. By the uniqueness theorem, $f$ is constant on $\Bbb D$. Is this correct? Let me know if there is any fallacy. Let $R$ be a commutative ring with 1, we define  $$N(R):=\{ a\in R \mid  \exists k\in \mathbb{N}:a^k=0\}$$  and  $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$  Could anyone help me prove that if $a\in N(R) \Rightarrow 1+a\in U(R)$?  I've been trying to construst a $b$ such that $ab=1$ rather than doing it by contradiction, as I don't see how you could go about doing that. How do I show that:  $$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$  This is actually problem B $4371$ given at this . Looks like a very interesting problem.   My attempts: Well, I have been thinking about this for the whole day, and I have got some insights. I don't believe my insights will lead me to a $\text{complete}$ solution.   First, I wrote $\sin\frac{5\pi}{14}$ as $\sin\frac{9 \pi}{14}$ so that if I put $A = \frac{\pi}{14}$ so that the given equation becomes, $$\frac{1}{\sin^{2}{A}} + \frac{1}{\sin^{2}{3A}} + \frac{1}{\sin^{2}{9A}} =24$$ Then I tried working with this by taking $\text{lcm}$ and multiplying and doing something, which appeared futile. Next, I actually didn't work it out, but I think we have to look for a equation which has roots as $\sin$ and then use $\text{sum of roots}$ formulas to get $24$. I think I haven't explained this clearly.         $\text{Thirdly, is there a trick proving such type of identities using Gauss sums ?}$ One post related to this is:  I don't know how this will help as I haven't studied anything yet regarding Gauss sums. In D&amp;D 5e &quot;&quot; is a condition that can affect a creature during the first round of combat, only until the end of their turn:  [... T]he GM compares the Dexterity (Stealth) checks of anyone Hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter. If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. A member of a group can be surprised even if the other members aren’t. - PHB 189  Does the action that causes the surprise count as part of the round, though? Say Renee, a rogue, achieves a stealth check of 22, easily beating an orc sentry's passive perception. She sneaks directly behind him and stabs him in the back. Surely the orc isn't surprised at all until the dagger hits him (or if Renee misses, she fumbles around behind him or something). What I think it really boils down to in the end is: when is initiative rolled? If (in our example) initiative is rolled before Renee's attack hits (or misses) how can we justify the orc recovering from his surprise before the action that triggered said surprise even occurs, should he roll better than Renee on his initiative? I'd like to go with initiative is rolled immediately following whatever event began the encounter (which could be an attack, but is likely to often be something else wherever surprise isn't intended), but I can't find any documentation to support this stance.  Why Does Anyone Care? There are many reasons you might want someone to be surprised while you're attacking - in this example if Renee is over L3 (and took the Assassin archetype) she'll get 'Assassinate' if the orc is surprised, and 'Death Strike' if she's over L17. That's a lot of potential extra damage!  that if the orc is surprised, Renee will be able to retreat and hide, without having to take the disengage action as her bonus action to avoid an attack of opportunity. Of course, there's no way for her to know whether the orc is surprised or not at that stage, so trying this would be a bit of a gamble... Choosing colors appropriately for red-green colorblindness I want a green for "correct answer" and a red for "wrong answer". My current choice is rgb(144, 238, 144) for the green and rgb(250, 128, 114) for the red. (See wikilearner.net/wiki for the colours in action.)  I have been told that these colors are very close in contrast, making it difficult for a person with red-green color-blindness. What would be a better choice? Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table I've done a truth table after reducing it to this and it seems to be equal to $\neg Q$:  $$\lnot Q \lor (\lnot Q \land R) = \lnot Q$$  But when I try to show it without a truth table (with just transformations), I end up in a loop between that and:  $$\lnot Q \land (\lnot Q \lor R)$$  Is there a way to show this is true without using a truth table? What am I missing?!  Thanks in advance! User logout on browser / tab close How to modify Drupal user's cookie so the cookie is not available anymore after a predefined time or when the user close his web browser or close all of the websites pages open in tabs ?  After some online searches i found that by setting the cookie lifetime to 0 should create temporary cookies that would be deleted once the browser is closed:  I first tried by modifing some ini_set() values in the settings.php :  ini_set('session.cookie_lifetime',  0); ini_set('session.gc_maxlifetime',   0);   I created a php.ini file located at the drupal site root, containing:  session.cookie_lifetime = 0;   But it does not work on my test site. Any ideas? What is the difference between a linear and a circular polarizer? When I bought my polarizer, a friend told me that I should get a circular one, because the linear ones can mess with the autofocus. Is this true? What should each be used for? How do constant aperture zoom lenses work? In Section 9 of Chapter 6, Lang constructrs for the first time a non-abelian extension as a splitting field $K$ of $X^n - a$ when the ground field $k$ does not contain primitive $n$-th roots of unity $\zeta_n$. He constructs an injective homomorphism of the Galois group $G(K/k)$ into the matrices of type   $$ M=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix} \quad \hbox{with}\ b\in \mathbb{Z}/n\mathbb{Z} \quad \hbox{and} \quad d\in (\mathbb{Z}/n\mathbb{Z})^*,$$  where $(\mathbb{Z}/n\mathbb{Z})^*$ the multiplicative group of integers modulo n.  Afterwards, he states in Theorem 9.4 that if $n$ is an odd positive integer prime to the characterstic and $[k(\zeta_n):k]=\varphi(n)$ (i.e, the primitive roots of unity are not in the ground field), then the above homomorphism is, in fact, an isomorphism. Besides, the commutator group is $G(K/k(\zeta_n))$, so $k(\zeta_n)$ is the maximal abelian subextension of $K$.  Lang proves the theorem inductively over the number of prime factors in the decomposition of $n$ (i.e. if $n=9= 3\cdot 3$, then the proof "would take two steps" to complete). The induction start with $n$ prime is written clearly and I agree with it.   Now come the questions:   I do not understand Lang's argument to prove that the commutator subgroup is the subgroup $\begin{pmatrix}1 &amp; 0 \\ b &amp; 1\end{pmatrix}$, $b\in \mathbb{Z}/n\mathbb{Z}$. What is this projection on the diagonal and how does one see that the factor group of the commutator group is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$?   By the way, direct computation of $x\, y\, x^{-1}\, y^{-1}$ for two arbitrary matrices $x=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix}$, $y=\begin{pmatrix} 1 &amp; 0 \\ c &amp; e \end{pmatrix}$ yields a matrix $\begin{pmatrix} 1 &amp; 0 \\ b(1-e)+c(d-1) &amp; 1 \end{pmatrix}$. As $n$ is odd, $2 \in (\mathbb{Z}/n\mathbb{Z})^*$, so taking $e=1$ and $d=2$, we indeed reconstruct the above-mentioned group.   The second (inductive) part of the proof is a complete mystery to me with virtually every sentence being a Theorem which needs an extra proof. Can someone give me some hints?   2.a. Lang writes $n=p\,m$ and takes $\alpha$ to be the root of $X^n-a$. It is clear that $\beta=\alpha^p$ is a root of $X^m -a$. Now comes the argument "By induction we can apply the theorem to $X^m-a$". I agree that we can apply it here but the result is that $K=k(\beta,\zeta_m)$ is the splitting field of $X^m-a$. In the diagram below, there appears $k(\beta,\zeta_n)$ out of nowhere.   2.b. Now Lang proceeds to apply the theorem on $X^p-\beta$ over $k(\beta)$. Bergman's "Companion" reminds to check that $[k(\beta,\zeta_p):k(\beta)]=\varphi(p)$ before applying the theorem and gives a hint "$[k(\zeta_p):k]$ and $[k(\beta):k]$ are relatively prime". I disagree with this hint. Take $n=21= m p$ where $m=3$ and $p=7$. Then $[k(\zeta_p):k]=\varphi(p)=p-1=6$ but $[k(\beta):k]=m=3$ and both are not co-prime. Can we apply the theorem here at all? Or if so what is the argument? If Bergman's argument is valid, where is the fault in my logic?  2.c. Even assuming that we can apply the theorem in 2.b., then we can say something about $k(\alpha, \zeta_p)$ and not $k(\alpha,\zeta_n)$. How do we proceed to checking everything Lang states in the last section of the proof (basically, everything after the diagram)?  EDIT: I have found the essentially same question How do I see all foreign keys to a table or column? In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL.