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I would like to compute the following probability

$$ P\left( ax \geq \sum_{i=1}^n b_i y_i \right) $$

where $a, b_i$ are constant coefficients (in my case, they are positive too) and $x, y_i$ are independent complex gaussian random variables with mean $0$ and variance $1$.

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Do you really mean complex Gaussians? If so, mean, variance do not give full specification. –  André Nicolas Jul 28 '11 at 19:40
    
Could you mean circularly symmetric complex Gaussian, and be interested in the behaviour of norms, or maybe projections on the real axis? –  André Nicolas Jul 28 '11 at 23:36

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up vote 2 down vote accepted

Like André Nicolas, I do not understand your use of complex, especially with the $\ge$ symbol. So what follows ignores it. I will also use Capitals for random variables

$aX$ is a Gaussian random variable with mean $0$ and variance $a^2$.

Similarly $\sum_{i=1}^n b_i Y_i$ is a Gaussian random variable with mean $0$ and variance $\sum_{i=1}^n b_i^2$ because of independence.

And $aX - \sum_{i=1}^n b_i Y_i$ is a Gaussian random variable with mean $0$ and variance $a^2+\sum_{i=1}^n b_i^2$ because of independence.

So $\Pr( aX \geq \sum_{i=1}^n b_i Y_i ) = \frac{1}{2}$.

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I am really sorry, I don't know what I was thinking. I will edit the question in a while. –  ACAC Jul 28 '11 at 23:41

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