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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27 views

Cosine of a triangular random variate

Good morning, I want to calculate the probability density function of a random variate $Z=cos(Y)$, where $Y=Φ_1−Φ_2$ and $Φ_{1,2}∼U(0,2π)$, that is both variables are uniformly distributed in $(0,2π)$...
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26 views

For $X$ exponential with mean $\frac{1}{\lambda}$, find pdf of $X^2, X^3,$ and $e^{-\lambda X}$

Supposed to express the answers in terms of $\lambda$. I tried X^2 and did $F_Y(x)=P[Y \leqslant X]=P[X^2 \leqslant x]=P[-\sqrt{x} \leqslant \sqrt{x}]$ Then this equals $F_X(\sqrt{x})-F_X(-\sqrt{x})$ ...
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57 views

How is double integral variable substitution different from one variable trigonometric substitution?

I'm studying variable change in double integrals and I understood the reasoning behind the formulas as described really well here. However, geometric arguments for analysis don't convince, as well as ...
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1answer
15 views

Determining the distribution of univariate transformation

If Y is uniformly distributed on the interval $(0, 1)$ and if $Z = –a * ln(1 – Y)$ for some $a > 0$, then to which of the following families of distributions does Z belong? Lognormal ...
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20 views

Probability of no heads in terms of a moment generating function

Define a R.V. $N \geq 0$, and let $M_N(s)$ be the associated transform. Assume that we have an unfair coin which lands heads with probability p and we toss it N times independently. Show that the ...
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1answer
42 views

Can the equation $\cfrac{n^2(s-2)-n(s-4)}{2}=2^p-1$ be transformed into an equation of the form $x^2 + D=AB^y$, for fixed $s$, $s \ge 3$

Can the equation $\cfrac{n^2(s-2)-n(s-4)}{2}=2^p-1$ be transformed into an equation of the form $x^2 + D=AB^y$ (where $D,A,B$ are fixed and $x,y$ are variables) when the variable $s$ is fixed to any ...
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1answer
63 views

Linear Transformation: Orthogonal Projections

Define $\mathbf{u_1} =$ $\begin{align} \begin{bmatrix} 0 \\ 0 \\ 1 \\1\\ \end{bmatrix}\end{align}$ and $\mathbf{u_2} =$ $\begin{align} \begin{bmatrix} ...
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1answer
67 views

Steps of transformation

Given the function $y=-5-3 \sqrt{-2x-4}$ and base function y= $\sqrt{x}$ describe the transformations that have been applied to obtain the function from the base function. I tried, horizontal ...
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38 views

matrix transformation : shear, composition

Suppose T_m is the shear in the x direction with factor k, and suppose T_p is the rotation by angle \theta use matrix multiplication to find the image of the vector v=(1;2) under the composition T_p ...
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2answers
36 views

Does translating a function change its domain?

In Spivak's Calculus chapter 3, there is a part which essentially states: $\textrm{if} ~~~ r(x)=x^2\ \textrm{such that} \ -17\leq x\leq \frac{\pi}{3}\\ \textrm{then} ~~~ r(x+1)=x^2+2x+1=r(x)+2x+1\ \...
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41 views

A question concerning Jacobians of coordinate transformation

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I ...
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1answer
18 views

Extracting sign of scaling from modelView matrix

I want to retrieve the sign of scaling for each axis from modelview matrix. Right now I am able to extract the sign only if all 3 signs are same but it fails when one of them is different. Here is the ...
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1answer
19 views

Transformation of functions: proof for the time period

Given the standard form of a trigonometric function: $a \times \cos(b(t+c)) + d$, what is the proof that the period $p = \frac{2 \times \pi}{b}$. We don't have the proof in our syllabus. I'm just ...
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18 views

Determine a change of variables to transform one DE to another

Given two ODE's, is it possible to determine if one can be obtained from the other via a change of variables? In particular, I have the two ODEs: $$ \begin{split} \frac{d^2y}{dt^2}&=\frac{8t-3}{...
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2answers
18 views

Find coordinates of intersection

The question says "The line with equation $y = - \sqrt{3}$ intersects the graph at points A and B, find coordinates of point B." I worked out that the graph formula is $y = 2\cos(2x)$ and I think I'm ...
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1answer
12 views

Can a boolean value concerning changing the sign or not of a scalar value considered a 1D rotation?

So going descending order on dimensions: 3D: 3 scaling scalars, 3 rotation scalars, 3 translation scalars 2D: 2 scaling scalars, 1 rotation scalars, 2 translation scalars 1D: 1 scaling scalar, 1 ...
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2answers
42 views

How to transform a 2-d space on a circle to a higher dimension space

I have 2 points $A=(x_A,y_A), B=(x_B,y_B)$ on a unit circle $O$. The distance between $A$ and $B$ goes through the perimeter of the circle. How can I transform this space to a space with higher ...
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32 views

Exponential to Weibull Distribution transformation

Let X~Exp($\lambda$) and let Y=$\lambda$$X^{1/\gamma}$. Find and name the distribution Y. So considering the CDF of Y, I have that $F_Y(y)=1-e^{-\lambda(y/\lambda)^\gamma}$. This is looking very ...
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0answers
8 views

Exponential Distribution Transformaion

Let X~Exp($\lambda$) and let Y=$\lambda$$X^{1/\gamma}$. Find and name the distribution Y. So considering the CDF of Y, I have that $F_Y(y)=1-e^{-\lambda(y/\lambda)^\gamma}$. Now I think the next ...
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1answer
63 views

Mapping from the z-plane to the w-plane

I'm struggling with this question Show that the transformation w=z-1/z maps |z-1|=1 in the z-plane to |w|=|w-1| in the w-plane. Any help would be much appreciated. Thanks
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1answer
111 views

A Lipschitz transform maps measurable set to measurable

Prove that a Lipschitz transform $T: \mathbb{R}^n \to \mathbb{R}^n$ maps measurable set to measurable. Assume the only thing that we know about Lipschitz transform is that we can find $M>0$ such ...
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1answer
38 views

How do I solve this complex numbers problem: transformation from the z plane to the w plane?

The point $P$ represents a variable point $z = x + iy$ in an Argand diagram. The point $Q$ represents a variable point $w = u + iv$ in a second Argand diagram and $x$, $y$, $u$ and $v$ are real ...
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18 views

Linear transformations that preserve permutations of a vector.

Forgive me if this question sounds silly. Let $v$ be a $m \times 1$ vector and $P$ be a $m \times m$ permutation matrix. Can there be a transformation $T$ such that $\min_\limits{P\neq I}\|Tv-TPv\|$ ...
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1answer
28 views

How to solve this graphical function transfromation problem?

How to solve problems like this. I always face problem in solving function transformation related problems. Is there any good way to solve problems like this..
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22 views

Triple integration, Spherical coordinates

How do we get limit such as $0\le\theta\le\pi$, $0\le\phi\le2\pi$ in spherical coordinate system where $$x=r \sin\theta\cos\phi, y=r \sin\theta\sin\phi, z=r \cos\theta$$ Why is the $\theta$-limit $[0,...
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15 views

Similarity between two 2-D Transforms

How can I measure the similarity between two 2-D transforms? For instance, I would like to find how much similar is the 4x4 Hadamard Transform (H) with the 4x4 integer DCT (D), as used in H.264/AVC (a ...
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1answer
232 views

A linear transform of a closed set is closed

A linear transform of a closed set $E\subset \mathbb{R}^d \to \mathbb{R}^d$ is closed. I have seen a lot of similar questions here, but none of them exactly addresses the issue. Please if you find ...
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36 views

Can the z-transform/laplace transform be generalized?

The fourier transform is a special case of z/laplace where the countour being projected on is the unit circle. The ztransform/laplace transform then generalizes this to a circle of any radius. Is ...
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31 views

transformation of differential equation

1.$$ x^2(1+u)dx + x^3(1-u)du = 0 $$ 2.$$ \frac {1-u}{1+u}du + \frac {dx}{x} = 0 $$ I know from 1 to 2 the equation is divided by $x^3(1+u)$, but i don't understand how that happens,was reading ...
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1answer
34 views

Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
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1answer
31 views

Is there a concise, specific name for a transform that consists of rotate, scale and translate?

I'm working on software that involves transforming between different mapping coordinate systems. In one part of the maths/logic, I have to derive, then apply a transform between two cartesian ...
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32 views

Tranformation of random variables

Let $X$ have the p.d.f $f(x)= e^{-x}$, $ x > 0$, $0 $ otherwise,find the pdf of $Y = X^2$ and space range $Y$. I use the change of variable formula The inverse is $${x} = \sqrt{{y}} $$ The ...
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1answer
19 views

What is this method of scaling called? Can it be generalised?

Consider the problem of finding the values of $\alpha_1, \alpha_2, ..., \alpha_k$, subject to constraints, such that the following equation is satisfied \begin{equation} \alpha_1 x_1 + \alpha_2 x_2 + ...
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29 views

fourier transform of a function that is only partially known

I'm interested in periodicity of a function that is not known completely, one way to think about it: our function $x(t)$ is defined at discrete points $t_1, t_2, ...$ but the values at these points ...
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68 views

Find me a sigmoid function with fixed point at the point of inflection in the unit interval

I am interested in finding sigmoid (S-shaped) functions $f$ on the unit interval $[0,1]$ such that For some $a \in (0, 1)$, $f(a) = a$ and $f'(a^-) > 0$ and $f'(a^+) < 0$. $f(0) = 0$ and $f(1) ...
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2answers
27 views

Before and after a transformation apply another transformation and its inverse?

So I have something like $y(x) = x\cdot5$, so whatever $x$ is, it's being scaled up by $5$. But I could center that scaling to a different point reference other than $0$. For example $10$. So for ...
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1answer
15 views

Method for transforming one curve around another?

I'm working with a complex problem involving waveforms. Essentially I want to bend a given waveform around a circle. At it's most basic, I want to take one curve on a linear graph and map it onto a ...
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1answer
13 views

Z transformation with $k$ from non-zero

we known that $Z$ transformation of $f_k$ is defined as $$F(z)=\sum_{k=0}^{\infty}f_k z^k$$ My problem is if $k$ starts from $m$, where $m >0$, then $\sum_{k=m}^{\infty}f_{k+m} z^{k+m}$ is still ...
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120 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
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75 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
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25 views

Transformations of discrete random variable

I understand how to do transformations. Well, I thought I did. But I can't seem to comprehend how to do a discrete to discrete. Here is the question I am working with. (I am currently studying for an ...
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5answers
34 views

f(2x + 1) transformation

I was working on a problem that asked: If there is a vertical asymptote at $x = 5$ for $f(x)$, where is the vertical asymptote for $f(2x + 1)$? The correct answer is at $x = 2$, but this ...
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1answer
115 views

Linear Transformations finding matrix in respect to a basis and coordinate change matrix.

Define $T: Poly_2 \ to\ Poly_2$ by $$T(at^2+bt+c)=3ct^2 +2at-b$$ 1) Show that T is a linear transformation and give a matrix A for T with respect to the basis $B=\{t^2,t,1\}$. 2) Give a ...
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1answer
37 views

Find Fourier transform of triangular function based on a Fourier results of rectangular

I have a triangular pulse given by $$x\left(\frac{t}T\right) = \begin{cases} 1-\frac {|t|}T, & \text{if $T\ge t$} \\ 0, & \text{otherwise} \end{cases}$$ Given that $F\left(\operatorname{rect}...
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1answer
59 views

Order of Operations for Horizontal Transformations

We know that when we want to combine two horizontal transformations, specifically that of translating and stretching a function, we have to translate $f(x)$ first, and then afterwards stretch it. ...
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49 views

PDF for $\frac1a Uniform(−a,a)$

Problem: Let $X$ be $Uniform(−a,a)$ distrubuted. Calculate the PDF for $Z = \frac1a abs(X)$. Attempt: I think graphically here. $X$ is $U(-a,a)$ with $PDF = \frac1{2a}$ so $abs(X)$ is $U(0,a)$ with $...
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1answer
123 views

Show random variables are mutually independent

Let $X_1$,$X_2$, $X_3$ denote a random sample from the distribution having p.d.f $$f(x) = e^{-x},\,\, 0<x <\infty,$$ zero elsewhere. Show that $$y_1 = \frac{x_1}{x_1 + x_2} $$ $$y_2 = \frac{...
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1answer
80 views

Mellin transform of Gumbel distribution

The probability density function (PDF) of Gumbel distribution is given as: $$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x-\mu}{\beta }\right)+\frac{x-\mu}{\beta }\right)\right)}{\beta }...
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2answers
52 views

Using the dimension formula to prove isomorphism

Let V be a finite-dimensional vector space and $T: V\rightarrow V $. T is a linear transformation. Use the dimension formula to prove that if T is injective, it must also be surjective; if T is ...
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124 views

Normalize a diagonal matrix such that each element belongs to $[0,1]$.

Let $\mathbf{x}\in\Bbb{R}^n$ be an $n$-dimensional real vector and $C\in\Bbb{R}^{n\times n}$ be a diagonal real matrix. Suppose that two vectors $\mathbf{r}_\min,\mathbf{r}_\max\in\Bbb{R}^n$ ...