Show that setting atcholsigma leads to a simulation from, Basic Statistics

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Show that setting atcholsigma leads to a simulation from

Given a 3 × 3 matrix Sigma:

a. Show that Sigma=cov(matrix(rnorm(30),nrow=10)) defines a proper covariance matrix.

b. Show that setting A=t(chol(Sigma)) leads to a simulation from N_p(0, Σ) by using the command x=A%*%rnorm(3).

c. Compare the execution times of this approach and rmnorm when simulating one vector and 100 vectors.

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