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idomiralin
 Post subject: Simulation of a random variable with the composition method
PostPosted: Wed, 8 Feb 2012 12:42:31 UTC 
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Hello!

I really need your help!

I have to simulate a random variable, which distribution function

F(x) = integral from zero to infinity x^y * e ^ -y dy, 0 <= x <= 1

with the composition method.

I think that F(x) must be divided in two known distributions and I don't know how to do it.

Please help me, because otherwise I don't have the right to go to the exam.

Thank you!
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outermeasure
 Post subject: Re: Simulation of a random variable with the composition met
PostPosted: Wed, 8 Feb 2012 15:57:01 UTC 
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idomiralin wrote:
Hello!

I really need your help!

I have to simulate a random variable, which distribution function

F(x) = integral from zero to infinity x^y * e ^ -y dy, 0 <= x <= 1

with the composition method.

I think that F(x) must be divided in two known distributions and I don't know how to do it.

Please help me, because otherwise I don't have the right to go to the exam.

Thank you!


You have \displaystyle F(x)=\int_0^\infty x^y e^{-y}\,\mathrm{d}y = \frac{1}{1-\log x}, so ...

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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