Question

W₁ = Y₁ + Y₂ + Y₃

W₂ = Y₁ - Y₂
W₃ = Y₁ - Y₂ - Y₃
(a) State the above in matrix notation.
(b) Find the expectation of the random vector W.

Answer 1

In matrix notation, W₁ = Y₁ + Y₂ + Y₃, W₂ = Y₁ - Y₂, and W₃ = Y₁ - Y₂ - Y₃. To find the expectation of W, we need the expectations of Y₁, Y₂, and Y₃. The expectation of W can be obtained by adding the expectations of the corresponding elements of W.

Step-by-step explanation:

(a) In matrix notation, the given equations can be written as:

W₁ = Y₁ + Y₂ + Y₃

W₂ = Y₁ - Y₂

W₃ = Y₁ - Y₂ - Y₃

(b) To find the expectation of the random vector W, we need the expectation of each element of W. Let's assume the random variables Y₁, Y₂, and Y₃ have expectations E(Y₁), E(Y₂), and E(Y₃) respectively. Then the expectation of W is:

E(W) = [E(Y₁ + Y₂ + Y₃), E(Y₁ - Y₂), E(Y₁ - Y₂ - Y₃)]

Since the expectation of a sum is the sum of expectations, we can simplify it as:

E(W) = [E(Y₁) + E(Y₂) + E(Y₃), E(Y₁) - E(Y₂), E(Y₁) - E(Y₂) - E(Y₃)]