# Tagged Questions

Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), (rigid-...

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### Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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### Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
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### Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
Let's say I have a canonical coordinate system in $\mathbb{R}^2$ described by the basis $\{e_1=\begin{bmatrix}0\\1\end{bmatrix}, e_2=\begin{bmatrix}1\\0\end{bmatrix}\}$. In it I have a vector $\vec{x}... 1answer 50 views ### What is the domain for which this integral transform is defined? Let$s=\sigma+it$the complex variable where thus$i^2 =-1$, and$\sigma$and$t$are real numbers. Let$\mu(k)$the Möbius function. It is possible determine the set of functions such that $$M \... 2answers 45 views ### translation and rotation of a parabola I am trying to translate a parabola to the origin, rotate by T radians and then translate back to the original position. I can calculate the new X and Y vectors using matrix operations and the regress ... 2answers 406 views ### What is the sense of bottom row of affine transform matrix? Usually affine transform matrix (in 2D) is represented like where block A is responsible for linear transformation (no translation) and block ... 1answer 560 views ### Understanding perspective transform matrix elements interpretation I am representing 3D points (vectors) in the following way: ... 1answer 67 views ### How do I draw this picture in squares of discrete \sqrt{z}? From Richard Kenyon's homepage gallery: I want to understand the mathematics of this, and similar/related transformations. ... An explanation in words (1st year uni level maths) would be ideal. I'... 0answers 9 views ### Probability density transformation for non-invertible mapping I am looking for a generalization of the result which states that the density of the sum of two random variables is the convolution of their densities. Specifically, if I have Z=f(X,Y), where p_{X,... 1answer 23 views ### matrix transformation of deformed rectangle I am working on touch screen calibration, and have come across a problem. My area of touch screen input is a Trapezoid which looks like a square on one side and a triangle on the other. (the angle ... 1answer 54 views ### Looking for the name of a formula [closed] Probability subject. The question is: If fx(x)=xe^(-x^2/2) for x>0 and Y=lnX find the density function for Y The solution is:(e)^(2y-1/2e^2y) I'm stuck on the part of the solution that uses this ... 0answers 27 views ### Proving non-linear mapping is invertible using partial derivatives only Given f : \mathbb{R} \rightarrow \mathbb{R}, it's possible to show that f is a bijection by considering its derivatives only: if the derivative is always positive or always negative, then the ... 1answer 254 views ### Z transform piecewise function I have this piecewise function:$$x(n)= \left\{ \begin{array}{lcc} 1 & 0 \leq n \leq m \\ \\ 0, &\mbox{ for the rest} \\ \\ \end{array} ... 1answer 26 views ### Problems again with an isomorphism Let$X$and$Y$be arbitrary sets and$f:X\rightarrow Y$an isomorphism. Prove that there exist a transformation$g:Y\rightarrow X$such that$f\circ g$is the identity in$Y$. I can't start the ... 1answer 74 views ### Active and passive transformations in Linear Algebra I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ... 4answers 62 views ### The formula for 3D rotation of the perspective of an image in 2D space Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of ... 1answer 31 views ### Find the marginal distribution of$V=X-Y$Problem: Show that the marginal density function of$f_V(v)$if$V=X-Y $is $$f_{V}(v)= \frac{1}{(1+|v|)^2}$$ for$ -\infty < v < \infty $. When the bivariate density function$f_{X,Y}(x,y)$is ... 0answers 24 views ### rotating a point using a previously rotated one I want to rotate a shape in an n dimensional space (n>3) around (about) the origin. knowing the outcome of rotation on a point like A, which is A', how can I find the rotation outcome on a point like ... 1answer 26 views ### Converting coordinates I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ... 3answers 42 views ### How to tansform${\sqrt{n-1}} + {\sqrt{n+1}} = q$into$q^4 - 4q^2n + 4 = 0$? [closed] Please help me tansform${\sqrt{n-1}} + {\sqrt{n+1}} = q$into$q^4 - 4q^2n + 4 = 0$? 1answer 104 views ### Transformation Matrix for cube in 2D My task is to transform the cube from the left corner to the big cube in the middle: What I did was: First i scale the cube: $$\begin{pmatrix} 4 & 0 & 0 \\ 0 & ... 0answers 19 views ### Gaussian function at a rotated and translated coordinate system I'm reading a paper and this coordinate transformation came along. In the z_{i}=0 plane the electric field is writen as E=\exp[-x_{i}^2]. The author says it's more convenient to work with the ... 1answer 33 views ### Mathematics of transformation of 2-D to 1-D coordination. Let's see an example In cartesian coordinate system: ... 2answers 26 views ### Transformation of a plane I have the (x,y)-plane$$\left\{(x,y,z)\in \mathbb{R}^3 | x,y\in \mathbb{R}, z = 0 \right\}.$$I need a transformation matrix to transform this to the plane$$ \left\{ (x,y,z) \in \mathbb{R}^3 | x+... 0answers 22 views ### Transformation of the gradient For a function$f\in C^2$,$f:\mathbb{R}^n\to\mathbb{R}$and a point$x\in\mathbb{R}^n$with$\nabla^2f(x)$positive definit one can calculate the new point$x^+=x+s$as follows: Change the ... 3answers 2k views ### How to find the equation of the graph reflected about a line? Consider the graph of$y = e^x$(a) Find the equation of the graph that results from reflecting about the line$y = 4$. (b) Find the equation of the graph that results from reflecting about ... 1answer 28 views ### How can I calculate the angle of a line/vector if the center of the image is not (0,0)? Simple image about the problem How can I calculate the alpha? My center of the image is (320,240) because it is a 640x480 image and the upper left corner is the (0,0). I tried to calculate it with ... 1answer 6k views ### Matrix for rotation around a vector I'm trying to figure out the general form for the matrix (let's say in$\mathbb R^3$for simplicity) of a rotation of$\theta$around an arbitrary vector$v$passing through the origin (look towards ... 2answers 40 views ### Finding the function of a sine graph that has both translation and transformation I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ... 1answer 61 views ### 2D Fourier transform of characteristic function of stripe on xy plane Given a stripe$X$on the xy-plane, namely$X\subset\mathbb{R}^2$, with$X=\{(x,y)\,|\; mx-\frac{1}{2}t \le y \le mx + \frac{1}{2}t$} and its "characteristic" function $$f(x,y) = \begin{cases} 1, ... 1answer 34 views ### Find functions F(\mathbf{x}) invariant under a map \mathbf{x} \to \mathbf{x'} We introduce a map \mathbf{x} \to \mathbf{x'}, defined as (for example on \mathbb{R}^3):$$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$Note that f,g,h are not all linear (or at least, I'm not ... 0answers 30 views ### How to prove this identity? Transformation theorem Let A\in\mathbb R^n be a measurable set with finite measure. For a fixed vector p\in\mathbb R^{n+1} define a cone with basis A and peak p as$$K(A,p)=\{tp+(1-t)q \in\mathbb R^{n+1} \,| \, q \... 2answers 33 views ### Transformation of a sphere and computing an integral by using sphere coordinates Let$V \subset\mathbb R^3$be the ellipsoid $$9x^2+4y^2+z^2≤36.$$ How can I express$V$as a transformation of a sphere and how can I compute the sphere $$\int_v x^2\,d\lambda^3(x,y,z)$$ with sphere ... 1answer 30 views ### Coordinate transform a triangle There is a triangle with points P1(x1,y1),P2(x2,y2),P3(x3,y3) on an XY plane. The final ... 1answer 48 views ### Transformation matrix For$x \in \mathbb{C}$define$A,B \in M(3\times3, \mathbb{C})as $$A = \begin{pmatrix} x & 0 & 0 \\ 0 & x & 1 \\ 0 & 0 & x \end{pmatrix}$$ and $$B = \begin{pmatrix} x & ... 2answers 43 views ### Bilinear transformation which maps z=(\infty, i, 0) and w= (-1, -i, 1) I have three equations after simplifying this a bit a+c=0 ai+b-c=0 b-d=0 How do I proceed further? If you care to know this is from the chapter Complex Variables 1answer 44 views ### Is there a universal symbol for transformation or operation? If I'm talking about a bunch of transformations (translation, rotation, scale, skew, etc) and I want to state A [some transformation] = B, what would be the symbol for [some transformation] or ... 2answers 48 views ### What is the difference between and projection and a reflection, in vector transformation? In my text book I have the problems of finding the standard matrix of the given linear transformation from \mathbb{R}^2 to \mathbb{R}^2; Projection onto the line y = -x. Reflection in the line ... 1answer 1k views ### Fourier Transform of a Polynomial Lets say you are given $$f(x)=1+x^3$$ and the definition of Fourier transform: \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\... 1answer 17 views ### Derive the length of the longest line segment that can be enclosed inside the region A. Q. Let A be the region in the xy-plane given by A={(x,y): x=u+v, y=v, u^2+v^2≤1}. Derive the length of the longest line segment that can be enclosed inside region A. My attempt: I found the equation ... 1answer 1k views ### Finding reflection of a matrix To 2 decimal places, what is the value of the lower-right entry in the reflection matrix Q_a if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook 3answers 31k views ### Image and Kernel of a Matrix Transformation So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation T: \mathbb R^5 \to \mathbb R^4 defined by T( \overrightarrow{x} )=A\overrightarrow{x}, ... 1answer 30 views ### Distribution of a transform of bivariate to univariate random variable? Suppose we have two random variables$$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$and a third random variable$$X=g(R,\Theta)=R\cos\Theta.$$What is the density p_X(x)? The ... 1answer 27 views ### Cone under similarity transformation Suppose we have a cone passing through the origin of xyz coordinate system. Now, the question is that whether we can find an invertible transformation on this coordinate system that turns the cone ... 1answer 25 views ### Image under T_j of the basis vectors e_1 and e_2. Define the linear transformation. Decide which of the mappings of \mathbb R^2 to itself given below are linear.$$\begin{align}T_1(x,y)&=(x+2y,y-2x)&T_2(x,y)&=(x,2x+y)\\T_3(x,y)&... 1answer 17 views ### inverse Mapping in Transformation of a random variable I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inferenceg^{-1}(A) = \{ x \in \chi : g(x) \in A\}$I know the idea that they want to ... 0answers 22 views ### Determinant in the transformation theorem Where does the$|det|$come from in the transformation theorem? It is pretty much the first time I saw a$|det|$in analysis. 0answers 19 views ### Rotation of a coordinate system Suppose that I rotate the (traditional) coordinate system$(x,y)$by an angle$\theta$to obtain a new coordinate system$(s,n)\$. Consider a velocity vector $$v = (v_x,v_y) = v_xe_x + v_y e_y,$$ where ...
$$f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right.$$ ...