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BookInquiry
 Post subject: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Tue, 17 Jul 2012 06:06:21 UTC 
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Where can I find out (the prove of)this topic?
Any books/ internet materials mention about this topic?

Xi has binomial distribution B(n, p)
Yj has binomial distribution B(m, p)

Xi+Yj has binomial distribution B(n+m, p)

P(\mu)
P(\lambda)

P(\mu+\lambda)
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Shadow
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Tue, 17 Jul 2012 06:09:32 UTC 
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BookInquiry wrote:
Where can I find out (the prove of)this topic?
Any books/ internet materials mention about this topic?

Xi has binomial distribution B(n, p)
Yj has binomial distribution B(m, p)

Xi+Yj has binomial distribution B(n+m, p)

P(\mu)
P(\lambda)

P(\mu+\lambda)


Just use the cdf function.

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outermeasure
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Tue, 17 Jul 2012 06:58:47 UTC 
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Shadow wrote:
Just use the cdf function.


... assuming independence!

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\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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Shadow
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Tue, 17 Jul 2012 07:00:08 UTC 
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outermeasure wrote:
Shadow wrote:
Just use the cdf function.


... assuming independence!


Yes, of course. I really should have said that. X(

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BookInquiry
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Wed, 18 Jul 2012 07:28:01 UTC 
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Yeah... random variables are assumed to be independent.
I don't even know the start of these proofs.
Any books/ internet materials should I refer to?
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outermeasure
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Wed, 18 Jul 2012 13:44:16 UTC 
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BookInquiry wrote:
Yeah... random variables are assumed to be independent.
I don't even know the start of these proofs.
Any books/ internet materials should I refer to?


Just compute the probability mass function.

\mathbb{P}(X+Y=r)=\sum_k\mathbb{P}(X=k,Y=r-k)=\dots

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\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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Shadow
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Wed, 18 Jul 2012 18:52:02 UTC 
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BookInquiry wrote:
Yeah... random variables are assumed to be independent.
I don't even know the start of these proofs.
Any books/ internet materials should I refer to?


Have you ever seen a "what is the distribution of the sum of these two random variables" problem before? They almost invariably go the same way each time where you do convolution to get the distribution of the sum. If you're having issues with this, check out this basic primer link.

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BookInquiry
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Fri, 20 Jul 2012 03:10:43 UTC 
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Shadow wrote:
Have you ever seen a "what is the distribution of the sum of these two random variables" problem before? They almost invariably go the same way each time where you do convolution to get the distribution of the sum. If you're having issues with this, check out this basic primer link.


I come across with joint probability distribution problems in this book: Advanced Mathematics 2 Perkins
It mentions the conclusions above without proofs.
I have no experience in these problems.

Shadow, Are you familiar with this doc?basic primer link
I don't even get the idea about it. Prerequisites are necessary?

I just learnt several distributions: Binomial, geometric, poison, normal,
I am expecting: sample distribution, joint probability distribution(linear/ non-linear combination of two or three sets of random variables), negative Binomial distribution... but I don't know about which references materials(books or internet) best learn about thee topics.
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Shadow
 Post subject: Re: Xi~B(n, p) Yj~B(m, p) Xi+Yj~B(n+m, p)
PostPosted: Fri, 20 Jul 2012 11:04:42 UTC 
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BookInquiry wrote:
Shadow wrote:
Have you ever seen a "what is the distribution of the sum of these two random variables" problem before? They almost invariably go the same way each time where you do convolution to get the distribution of the sum. If you're having issues with this, check out this basic primer link.


I come across with joint probability distribution problems in this book: Advanced Mathematics 2 Perkins
It mentions the conclusions above without proofs.
I have no experience in these problems.

Shadow, Are you familiar with this doc?basic primer link
I don't even get the idea about it. Prerequisites are necessary?

I just learnt several distributions: Binomial, geometric, poison, normal,
I am expecting: sample distribution, joint probability distribution(linear/ non-linear combination of two or three sets of random variables), negative Binomial distribution... but I don't know about which references materials(books or internet) best learn about thee topics.


I learned about the basic distributions from Sheldon Ross' basic book on the subject (he also has a more advanced one, which is certainly not appropriate if you have never seen convolution before).

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