Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] = ₁, that E[Y₁] = E[Y₂] = 2 and that var(X₁) = var (X₂) = var(Y₁) = var (Y₂) = o2 > 0. Let X = (X₁ + X2) and Y = (Y₁+Y2). Define a random variable S² via 2 2 82 = 1 2 Σ(X₁-X)² + Σ(Yi-Y)² i=1 i=1 The aim of this question is to describe the joint distribution of X - Y and S² (a) Let U be a 4 x 4 matrix whose first two rows are 1/√2 1/√2 0 0 (1/V/2 1/√2) 0 0 1/√2 1/√2) Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!] (Let W be the 4-dimensional random vector W = (X1, X2, Y₁, Y2), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S2 as a function of V3 and V4 and that X - Y is a function of V₁ and V₂. Use this to describe the joint distribution of X - Y and S². a): U=₁ OM OML OMLOMI- 。E。 ON TIN

2 Consider the matrix A = and the vector u = Let B = (A – A")2 and define vector v = Bu. If v = , write 1 down the value of vị. [Please enter your answer numerically in decimal format.

2 0 1 4 5 30x3 = 43 = (25² 0 ..] Can we use Cramer's Rule to check if the column vectors of A are linearly (in)dependent? Explain why or why not and regardless, identify all linearly independent column vectors in A. *Derive the vector x3 in Ax3 = b by two different methods (please show the essential calculations- outlining the method(s) isn't enough to get you credit).

Find the stable vector of this Matrix:  0 0 1 0 1/2 1/2 1/3 2/3 0