Notation: to consider a 'Gaussian' approximation of the d-dimensional unit sphere, by considering a Gaussian random vector X N (0, I/d), where I is the d d identity matrix. This means that for large input dimension d, a draw X from the N (0, I/d) Gaussian distribution will concentrate to the unit sphere X21.

If we draw two datapoints X, X' i.i.d. from N(0, I/d) where I is the d d identity matrix, show that<X,X>=O(1/d)with high probability using the Central Limit Theorem. In other words, show that X, X'is a random variable of zero mean and standard deviation proportional to1/d . Conclude that for a constant C > 0, XX(2+C/d)for large d with high probability.