Question

1.suppose that Y1 and Y2 are independent random variables

2.suppose that Y1 and Y2each have a mean of A and a variance of B

3.suppose X1 and X2 are related to Y1 and Y2 in the following way:

X1=C/D x Y1

X2= CY1+DY2

4.suppose A, B, C, and D are constants

What is the expected value of X1?

What is the expected value of X2?

What is the variance of X1?

Answer 1

Given that

Mean of Y₁ AND Y₂: E(Y₁) = E(Y₂) = A

Variance of Y₁ and Y₂: V(Y₁) = V(Y₂) = B

Covariance Y₁ and Y₂: Cov(Y₁,Y₂) = 0 (since they are independent)

X₁ = (C/D)*Y₁ and X₂ = CY₁ + DY₂

Now, some rules regarding exoected value and variance are:

E(k) = k

E(kX) = k*E(X)

E(kX + sY) = k*E(X) + s*E(Y)

V(k) = 0

V(kX) = k² * V(X)

where k,s = constant

(1) Expected value of X₁ = E(X₁) = E[(C/D)*Y₁] = (C/D) * E(Y₁) = (C/D) * A

So, E(X₁) = (C/D) * A

(2) Expected value of X₂ = E(X₂) = E(CY₁ + DY₂) = C * E(Y₁) + D * E(Y₂) = C * A + D * A = (C + D) * A

So, E(X₂) = (C + D) * A

(3) Variance of X₁ = V(X₁) = V[(C/D)*Y₁] = (C²/D²) * V(Y₁) = (C²/D²) * B

So, V(X₁) = (C²/D²) * B