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Let $X = A \cdot B$, where $A$ and $B$ are unit length vectors with $m$ elements, and no element of $A$ or $B$ is negative. What is the distribution of $X$?

If it helps, we can assume that the elements of $A$ and $B$ both started off as Gaussian with mean $0$, and a constant was added to each element so that no element is negative, prior to normalizing to unit length.

To be more specific, what I really want to know is the probability that $X$ is greater than some fixed threshold between zero and one.

Ken
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ken
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  • Do you mean $A$ and $B$ are drawn from this set?

    $$S = { x \in \mathbb R^m \ | \ x_i \geq 0, |x| = 1 }$$

    And if so, can you characterize the distribution more carefully? What for instance does mean of zero mean?

     – Simon S Jun 05 '15 at 16:14
  • Thank you for your question. It has been a long time since college, so I don't know for sure if I'm interpreting your set definition correctly.

    ||A|| = ||B|| = 1.

     – ken Jun 06 '15 at 15:04
  • each element of A and B are real numbers greater than or equal to 0. ||A|| = ||B|| = 1.

    A and B both represent real spectral intensities, e.g. Watts/Hz. It's not possible for these to have negative values. For similar calculations, It's not uncommon to mean-center these vectors, e.g. sum(A) = 0. ||A||=1.

    If you do that, it's not unreasonable by inspection of the typical data after setting the mean =0, to assume the elements of A are normally distributed around zero.

    But for the specific calculation I'm doing, I can't set sum(A)=0.

     – ken Jun 06 '15 at 15:18

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