Confusion about several different definitions of continuous random variables It seems like different books define continuous random variables differently.    pdf definition: The random variable X is continuous if a nonnegative function f exists, that is defined for all $x \in (-\infty, \infty)$ such that $P(X \in B) = \int_B f(x) dx$ (Sheldon Ross) cdf definition: The random variable X is continuous if its cdf is continuous and differentiable. Moreover, the derivative of the cdf is continuous except for a finite number of points. (Wackerly) Another cdf defintion: The random variable X is continuous if the cdf is a continuous function. (Wikipedia ) uncountable range:  The random variable X is continuous if its range is uncountable infinite/set of possible values is uncountable infinite. uncountable sample space:  The random variable X is continuous if the sample space is uncountable infinite.   My question are:   a) How do these definitions relate to each other? Are they equivalent? Why?  b) What is the difference between a continuous and an absolute continuous random variable?  c) Is it possible that the difference between continuous and discrete random variables vanishes from the point of view of measure theory? I have looked into several books about probability based on a rigid measure theoretic approach and none of them even mention the fact that there are different kinds of random variables.  d) In particular, is 4.) and 5.) equivalent? To me this doesnt seem to be necessarily the case...  e) Which is the "correct" definition from the above for a continuous random variable?  I would be happy also about answers from the point of view of measure theory, which probably answers this questions the most precisely.
"Absolutely continuous random variable" vs. "Continuous random variable"? In the book Limit Theorems of Probability Theory by Valentin V. Petrov, I saw a distinction between the definitions of a distribution being &quot;continuous&quot; and &quot;absolutely continuous&quot;, which is stated as follows:  &quot;...The distribution of the random variable $X$ is said to be continuous if $P\left(X \in B\right)=0$ for any finite or countable set $B$ of points of the real line. It is said to be absolutely continuous if $P\left(X \in B\right)=0$ for all Borel sets $B$ of Lebesgue measure zero...&quot;  The concept I am familiar with is:  &quot;If a random variable has a continuous Cumulative Distribution Function, then it is absolutely continuous.&quot;  My questions are: are the two descriptions about &quot;absolute continuity&quot; in (1) and (2) talking about the same thing? If yes, how can I translate one explanation into the other one?
Continuous weighted random distribution, biased toward one end I am currently contributing to a particle system for our game and developing some emitter shapes.  My uniform random distribution along a line or along a rectangular area works fine - no problem.  But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.  I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient. Capitalization of explanation of abbreviations Poisson Distribution of sum of two random independent variables $X$, $Y$ $X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to derive it. Evaluate the integral $\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x dxdy}{\sqrt{4a^2x^2-(x^2+y^2)^2}}$ Are there idioms or expressions in English that describe going from one bad situation to one that's even worse?  I heard "between a rock and hard place" but this describes a dilemma not really a transition. I am looking for an expression that describes someone trying to get out of a bad situation but after much effort, the situation only got worse. Group theoretic meaning of natural isomorphisms between certain functors So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to  $G$ specified by  $\forall g$ $\in G$ $\exists$   $\phi(1)=g$. Each of these homomorphisms is in 1-1 relation with the elements of $G$.  Consider groups for a moment in the context of category theory, viewing the groups as one object categories and thereof group homomorphism as functors. What would a natural isomorphism mean in group theory terms for G?  My thoughts are that the natural transformations equate to maps between homomorphisms and since the homomorphisms are in 1-1 relation with the elements of G, then a natural transformation is a map between group elements, i.e another element of the group. I'm not entirely convinced by this.   Anyway continuing my reasoning a natural isomorphism is a map between the homomorphisms that has an inverse, the thing is in relation to the elements of G this is every element? Furthermore I'm looking to find an equivalence class defined by this natural isomorphism, is this just a collection of the elements that are each others inverses?  Thanks in advance for any assistance. Can negative levels get rid of racial HD? What happens when one is affected by something such as energy drain, and permanently loses levels when one only has racial HD? Does the victim of the spell die? Or do they permanently lose racial HD? How to convert hex string to float in Java? How to "freeze" a cloth animation at X frame? How would one freeze a simulated body at frame X and use it as a static mesh?  Following a , I've been able to do a cloth animation to my model, now I'd like to have that made static, without the use of the timeline.   The thing is, I'm modelling something that's not going to be an animation, but a static object (something like a statue/figurine). So I don't need the animation, only the shape at a particular frame that I find convenient. I'd like eventually once this is done, to also modify that deformed grid.   I've tried to follow what was said , but couldn't figure out what chipmasque was saying. I'm quite new to blender (been using it on and off, just enough to forget everything).  I'm using blender 2.63, if it is of any relevance. Avoid footnotes from appearing before mark I have these \footnotetext and \footnotemark as illustrated in the figure below. I used these tags to enforce my own numbering (and because using only \footnote was not working the way I expected).    The code I use for generating the mark is:   \caption{Lista de notas presentes em um piano de 88 teclas,  com notação em Inglês, frequência em Hertz e comprimento de  onda em centímetros.\protect\footnotemark[1]}   Is there any way I can enforce my \footnotetext to appear in the same page as my \footnotemark? The latest beta update made the app crash a lot:   when I press 'Add comment'. after deleting a comment. after up/down/unvoting a post (happened when I upvoted an answer) after editing a post (but the edit is applied).   Spec: Nexus 5 on Lollipop 5.1.1, SE App v1.0.66 (beta) Does a correlation matrix of two variables always have the same eigenvectors? If $I_n=\int _0^{\pi }\sin^{2n}\theta \:d\theta $, show that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $  Hence calculate $\int _0^{\pi }\:\sin^4t\cos^6t\:dt$  I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, but I am not very good at English, what does it mean Hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $ do we need to prove this part as well or is it just a hint to use? and for the other calculation to find $\int _0^{\pi }\:\sin^4t\cos^6t\:dt$ is there something in the first part that I can do to help me solve this question because otherwise it becomes very long and it's part of the question. I'm trying to fill empty space in my mesh, and I looked at the knife tool page on the Blender Wiki, and I'm not sure if there's a way to make it so that a new edge is made using the knife tool. I took a screenshot, and hopefully it'll explain better than I can of what I'm trying to do. Oh and I'm using Blender version 2.71, by the way. python ImportError: No module named primes What is the word for when someone is overly nice and actually isn't nice at all? What is the word for when someone is overly nice and actually isn't nice at all because of the unnecessary added information/comments? For example, they will say things like "Your hair looks so good that wayyyy omg you should do it that way ALL the time! Looks way better than how you usually do it." And the word I'm looking for is not passive aggressive and it is not sarcasm. There is a word for these types of people. What is it? These people think they're being kind but actually blunt and rude. Another example would be for someone to say "hey you're beautiful, but maybe if you didn't wear harsh makeup you would look better" and they intend to sound like they're being nice when you know they don't like you and are saying in out of spite. Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $k$ is divisible by two, such as say $k=8$, or $k=12$.  If so, can they be constructed easily? I'd be grateful if you could provide a few examples. If I change the ISO settings on my camera, obviously the gain of the system is increased, amplifying the signal from the sensor. What's not clear to me is where the amplification takes place. I see several possibilities:   In the sensor, by increasing voltage or some other mechanism Via an analog amplifier outside of the sensor Digitally, after the signal has been digitized, but before storing data in the RAW file As a parameter applied solely in creating an image from RAW   If #4 is true, then you could take a 4-stop overexposed RAW picture at ISO 1600, and then in post processing produce a JPEG at ISO 100 that would be the same as if the original photo had been shot at ISO 100.  If #1 or #2 is true, then a RAW file shot at ISO 1600 would actually contain more information about shadows, and an ISO 100 RAW would contain more information about highlights. If the subject of a sentence is separated by "and/or", should the verb be pluralized, as with "and", or agree with the rightmost subject, as with "or"? For example:     His co-workers and/or his boss was at fault.   vs.     His co-workers and/or his boss were at fault. How to set mount point of boot partition to /boot/efi?