# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 }...
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 ...
### 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 $\mathbf{r}_\min,\mathbf{r}_\max\in\Bbb{R}^n$ ...