Currently very confused about lecture problems, please explain thoroughly in R programming.

Problem 1: Recall that a a-dimensional random variable X is and to be distributed normally Normal(m, E), where me Bf and I e Rod is a symmetric positive definite matrix, if its PDF is ghen by, /x(x) = - - 1((x - myS (x - mj) (i) Show that if m = 0 6 R and E = I (the identity matrix) then the PDF has the form, This Is called the "standand normal d-dimensional distribution". (i]) Suppose X has the standard normal d-dimensional distri- bution and that I is a symmetric positive definite matrix. Define the linear transformation y : " + R" given by the matrix multi- plication, a(x) = Ex. We can use this transformation g() to define the transformed random variable Y = #(X). Show that Y will be distributed as a Normal(0, 2) distribution. (Hint: You need to use the distribution transformation formula, namely, Do not forget that y() is a linear transformation of d variables, so its derivative will be a d x d matrix. (imi) Write your own script in R, do not use avrnorm, to draw 105 independent samples from the 3-dimensional normal distribution where m = [0,0, 0) and, 6 12 A E= 12 29 5 5 12 "To get started art A beatriz (NA, nrow = 146, ncol-3) to greer- Me an empty matrix. Now use more to fill each now with 10" independent standard normal samples. (Since we already know how to sample with the standard normal we can simply was the build-In command in R for doing it.) You now have a matrix whose cols consist of three-dimensional samples from the standard normal distribution. Take the linear transformation s(x] = Ex. Apply this linear trans- formation (which acts on 3-dimensional vectors] on each ood of this matrix. For example, If v is the flast call of the matrix . then We will replace this first cal by Dy (in other words, you set on this three dimensional vector v by applying the linear map g(v]). Do this for every single cal. You will now have a transformed three dimensional data set. Call this new matrix H. Each cal of this matrix will represent a three-dimensional sample. According to (li) of this exercise these three-dimensional samples will be nor- mally distributed with the cornet & parameter