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$n$. 1 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\$.