The vector of means m and the covariance matrix C_X of two jointly Gaussian random variables X1 and X2 are given below. =[11] and C_X =[4339]

X1 and X2 are transformed linearly into two new variables Y1 and Y2 according to ₁ = ₁ 2_{2 2} = 3₁ + 4₂ a) Find the means of Y₁ and Y₂. b) Find the covariance matrix C_Y of Y₁ and Y₂ using the formula for the linear transformation of Gaussian v.a.. Then determine (from C_Y) the variances of Y₁ and Y₂ and the covariance between Y₁ and Y₂. c) Using the property of variances VAR( ) = ²VAR() + COV(, ) check that the variance of Y1 is the same as found in b). d) Verify that if there is only one new variable Y₁, you also allows to find the variance of Y1, i.e., if T = [1 -2]. e) Determine the probability density function (pdf) of Y₁. f) Determine the joint pdf of Y₁ and Y₂.