Expression for The Variance of a Linear Combination - Statistics Assessment Answer

Statistics Assessment Task

Let X1 and X2 be a scalar random variable where X1 and X2 have the following covariance matrix

S= where 0 < ρ < 1, i.e. X1 and X2 are positively correlated.

Question 1

Write down an expression for the variance of a linear combination w₁x₁ + w₂x₂ in terms of w1, w2 and ρ.

Question 2

Suppose we want to ?nd w= (w₁, w₂)0 that gives a linear combination with maximal variance subject to the constraint that w₁² +w₂² =1. Derive ?rst order conditions that must be satis?ed for a solution to have maximal variance

Question 3

Derive a value of w that satis?es the ?rst order conditions given in your previous answer and leads to a linear combination with maximal variance

Question 4

Are there any other values of w that satisfy the ?rst order conditions given in your answer to Question 2? What can we say about these linear combinations?

Question 5

How do your answers to Question 3 and Question 4 change when X1 and X2 are negatively correlated?

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