# probability – How to prove that the projection length \$frac{x^T y}{|x|}\$ is normally distributed if \$x,ysim N(0,I_p)\$?

$$x,yinmathbb{R}^p$$ are independently drawn from the standard multivariate Gaussian distribution $$N(0,I_p)$$. How do you prove that the projection length $$z=frac{x^T y}{|x|}$$ is normally distributed?

If $$x$$ is fixed, then $$z$$ is just a linear combination of the entries of $$y$$, so it is normally distributed. And we can easily calculate the mean and variance, so $$z|xsim N(0,1)$$ for any $$x$$. Is this enough to conclude that $$zsim N(0,1)$$ unconditionally?