8.4（c） 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] = = μ₁, that E[Y₁] = E[Y₂] = 2 and that var(X₁) =var (X₂) = var (Y₁) = var (Y₂) = o² > 0.Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via1/12 (2² (x₁ - X)² + 2(x − 8)²)ΣΥ; -i=1i=1The 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(1/√21/√2)00 1/√2 1/√2Choose two further rows so that U is an orthogonal matrix. [ Hint: it's a good plan to useplenty of zeros!](b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y₂)T, and define V to be therandom vector V = UW. Find the mean vector and variance-covariance matrix of V.Show that you can write S2 as a function of V3 and V4 and that X -Y is a function of V₁and V₂. Use this to describe the joint distribution of X - Y and S².

To do this, we need to utilize the orthogonal matrix U and the transformation V=U W, where W is the…

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 Ỹ = (Y₁ + Y₂). Define a random variable S² via 2 2 Σ(x − x)2 + Σ - 1)2 - Σ(X-8²) i=1 i=1 The aim of this question is to describe the joint distribution of X - Y and S² 82 = (a) Let U be a 4 x 4 matrix whose first two rows are (1/V2 √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!] (b) 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. 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. =