This problem outlines a proof of the following equalities?

Question

if ˆX = aX + b and ˆY = cY + d, then a) pˆX,ˆY = pX,Y and b) Cov[ˆX,ˆY] = acCov[X,Y]

(a) Show that ˆX −E[ ˆX] = a(X −E [X]), ˆY −E[ ˆY ] = c(Y −E [Y ]).

(b) Use part (a) to show that Cov [ ˆX, ˆY ] = acCov [X,Y ] .

(c) Show that Var[ ˆX] = a2 Var[X] and Var[ ˆY ] = c2 Var[Y ].

(d) Combine parts (b) and (c) to relate ρ ˆX, ˆY and ρX,Y .

Answer 1

You are being asked to prove that covariance and correlation are invariant to a linear transformation.

Make use of the mathematical definitions of covariance, variance, correlation.and the rules of mathematical expectation (Have you guys derived these rules yet, or should I guide you to proofs of them?)

definitions, where A and B are continuous random variables

cov = E[(A - E(A)) * (B - E(B)]

var = E(A - E(B))^2, which reduces to E(A^2) - E(B)^2

correlation = cov/(sqrt(var(A) * sqrt(var(B))

rules of mathematical expectation:

1. The expectation of a constant is that constant, E(k) = k
2. The expectation of a sum is the sum of the expectations. E(A + B) = E(A) + E(B)
3. The expectation of a product is the product of the expectations. E(A * B) = E(A) * E(B)

You can use basic algebra, the appropriate mathematical definition, and the rules of expectation to solve each of the steps a, b, c, and d in the problem.

Best of luck! Please check back in if you run into difficulties.