3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x)...

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3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x) = = exp(-²/(r- (x − *) Σ-1(x – - â)). Here, & ER" is the mean of the distribution, and Σ => 0 is its covariance matrix, and the value C is such that [p(x) dx = 1. 1. (2 points) Show that when â = 0 and Σ = I, the constant C takes the value C = (2n)"/2. Hint: The integral of a product of functions fi, each depending on a single variable zi, can be expressed as a product of n one-dimensional integrals: n [ ÎI fi(zi) dz = Ï _ fi(ži) dzi. i=1 i=12₁ER Note that when calculating the individual integrals, you may use the 1-d result for standard normal distribution without proof. 2. (2 points) Using a spectral decomposition of Σ, show that in general, for arbitrary and matrix > 0, we have C = (2π)"/2√det Σ. Hint: for any integrable function f and square invertible matrix P, we have f(x) dx = |det P. S.cz, S(P2) dz. ZER" IER" 3. (2 points) Show that is indeed the mean of the distribution, that is: â = B(x) = √xp(x) dx. 4. (2 points) Show that is indeed the covariance matrix of the distribution, that is: (x-2)(xê) p(x) dx. Again you may assume that the result holds true when n = 1. Hint: use an affine change of variables in order to reduce the problem to that of computing the covariance of the distribution obtained upon setting 2 = 0 and Σ = I, which is the matrix Σ= M:= Rn (2m)n/² · ___²2¹e-1²³² d₂. 5. (2 points) Explain how to generate samples from the multivariate distribution, assuming that you have access to a sample generator for the normal distribution, which is the scalar distribution with zero mean and unit variance. ZER 3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x) = = exp(-²/(r- (x − *) Σ-1(x – - â)). Here, & ER" is the mean of the distribution, and Σ => 0 is its covariance matrix, and the value C is such that [p(x) dx = 1. 1. (2 points) Show that when â = 0 and Σ = I, the constant C takes the value C = (2n)"/2. Hint: The integral of a product of functions fi, each depending on a single variable zi, can be expressed as a product of n one-dimensional integrals: n [ ÎI fi(zi) dz = Ï _ fi(ži) dzi. i=1 i=12₁ER Note that when calculating the individual integrals, you may use the 1-d result for standard normal distribution without proof. 2. (2 points) Using a spectral decomposition of Σ, show that in general, for arbitrary and matrix > 0, we have C = (2π)"/2√det Σ. Hint: for any integrable function f and square invertible matrix P, we have f(x) dx = |det P. S.cz, S(P2) dz. ZER" IER" 3. (2 points) Show that is indeed the mean of the distribution, that is: â = B(x) = √xp(x) dx. 4. (2 points) Show that is indeed the covariance matrix of the distribution, that is: (x-2)(xê) p(x) dx. Again you may assume that the result holds true when n = 1. Hint: use an affine change of variables in order to reduce the problem to that of computing the covariance of the distribution obtained upon setting 2 = 0 and Σ = I, which is the matrix Σ= M:= Rn (2m)n/² · ___²2¹e-1²³² d₂. 5. (2 points) Explain how to generate samples from the multivariate distribution, assuming that you have access to a sample generator for the normal distribution, which is the scalar distribution with zero mean and unit variance. ZER