Just to be clear, if Z=XD1 plus we know that Z and D1 are independent then the correlation between Z and X is 0.872. This is because the covariance of Z and X is just the variance of X (which is 1) and the standard deviation is 1., and the variance of Z is the sum of the variances of X (1) and D1 (.316) (by independence) since .316^2 = .1. You can then calculate the correlation by taking 1/sqrt(1.316) = 0.872. So, as Phil Goetz said I don’t know what the comparison proves.
Edit: Removed useless latex and corrected a mistake.
Just to be clear, if Z=XD1 plus we know that Z and D1 are independent then the correlation between Z and X is 0.872. This is because the covariance of Z and X is just the variance of X (which is 1) and the standard deviation is 1., and the variance of Z is the sum of the variances of X (1) and D1 (.316) (by independence) since .316^2 = .1. You can then calculate the correlation by taking 1/sqrt(1.316) = 0.872. So, as Phil Goetz said I don’t know what the comparison proves.
Edit: Removed useless latex and corrected a mistake.