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I would like to calculate or at least to estimate some expectation.

Let $r_k$, $k=1,\ldots, 2n$ be random variables with $P(r_k=1)=P(r_k=-1)=\frac 12$ and such that half of them $r_k=1$ and half $r_k=-1$. Let $b_k$, $k=1,\ldots, 2n$ be real numbers.

I would like to calculate $$ E\left(\prod_{i=1}^n\prod_{j=n+1}^{2n} \exp(r_ib_ir_jb_j)\right). $$

Any ideas would be very helpful.

Thank you.

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You will need more information about the joint distribution of the $r_k$. For instance, suppose $n=2$ and $b_k = 1$ for $k=1,2,3,4$. One possible model is that you flip a coin; if it is heads you assign the $r_k$ the values (1,1,-1,-1) and if tails (-1,-1,1,1). Then your expectation is $e^{-4}$. Another possible model is that if the coin is heads you take the $r_k$ to be (1,-1,1,-1) and if tails (-1,1,-1,1). Then the expectation is 1. – Nate Eldredge Feb 22 '12 at 4:07
All I know that $r_k$ are Radamacher random variables... Maybe it is possible at least to estimate this expectation? – David Feb 22 '12 at 4:27
The spelling is Rademacher. Maybe you can find out something about the joint distributions. – Gerry Myerson Feb 22 '12 at 4:42
You'll have to be more specific about what you mean by "estimate". Perhaps some background on why you are interested in this problem would help. – Nate Eldredge Feb 22 '12 at 5:14
Here's a cautionary tale about thinking that the background of the problem wouldn't help :-) – joriki Feb 22 '12 at 13:35

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