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4.3 Probability and linear algebra 1. Let v1, v2 € Rd be two vectors sampled independently from the standard Gaussian distribution (which has mean vector 0 and covariance matrix equal to the identity). Compute the expected dot product between ₁ and v₂. 2. Compute the variance of the dot product. 3. Use Chebyshev's inequality to derive an upper bound on the probability that the dot product between ₁ and v2 is less than or equal to e, for arbitrary € > 0. This is meant to show that, for large enough d, randomly generated Gaussian vectors are approx- imately orthogonal. This is used in a dimensionality reduction method called random projection, which is justified by a result called the Johnson-Lindenstrauss lemma.