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 rmas September 7th, 2011 08:42 AM

Joint probability for an infinite number of random variables

Hi,

I have the following question :
How do we estimate the joint probability $Pr(X_1,... X_n)$ when $n \rightarrow \infty$ ?

Thanks a lot.

 Math Dreamer September 11th, 2011 05:57 AM

Re: Joint probability for an infinite number of random varia

Quote:
 Originally Posted by rmas Hi, I have the following question : How do we estimate the joint probability $Pr(X_1,... X_n)$ when $n \rightarrow \infty$ ? Thanks a lot.
Let each random variable be$X(i)=X_i(S)=\{x_1, x_2, \cdots, x_{|S|}\}$ where $I=\{1,2, \cdots, n\}$ and $S$ is sample space.

The product set $\displaystyle \prod_{i=1}^n X(i)=\{h\; |\; (\forall i\in I) h(i)\in X(i)\}$ where $h(i)$ is a joint distribution.

$h(i)$ is a set of $|S|$ number of $n$-tuples.

The probability with respect to each random variable $X(i)$ is $g(i)=\displaystyle \sum_{i=1}^n h(i)$

The probability = $p(S)=\displaystyle \sum_{i=1}^n \displaystyle \sum_{j=1}^n \displaystyle \sum_{k=1}^n\cdots \quad \displaystyle \sum_{z=1}^n\cdots h(i)\cdot h(j)\cdot h(k) \cdots\quad h(z)\cdots = 1$

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