I must be misunderstanding the question, but $^2$ <x, y>^2$ is a sum of the terms of the form $x_i x_j y_i y_j.$ The expectation of this term vanishes, unless $i=j,$ in which case it (the . expectation) is the square of the expectation of the square of the standard Gaussian. The square of the standard gaussian is the chi-square distribution with one degree of freedom, whose mean is $1,$ so the whole thing should be $n$.
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I must be misunderstanding the question, but $^2$ is a sum of the terms of the form $x_i x_j y_i y_j.$ The expectation of this term vanishes, unless $i=j,$ in which case it (the . expectation) is the square of the expectation of the square of the standard Gaussian. The square of the standard gaussian is the chi-square distribution with one degree of freedom, whose mean is $1,$ so the whole thing should be $n$. |
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