How Do You Prove the Variance Formula for a Linear Combination of Variables?

[cse63146]

Homework Statement

Let \Sigma =
( var(X1) cov(X1, X2) )
( cov (X2. X1) var(X2) )

Show that Var (a_1 X_1 + a_2 X_2) = a^T \Sigma a

where a^T = [a_1 a_2] is the transpose of the of the column vector a

Homework Equations

The Attempt at a Solution

I got this far:

Var (a_1 X_1 + a_2 X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + 2a_1 a_2 Cov (X_1, X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + a_1 a_2 Cov (X_1, X_2) + a_1 a_2 Cov (X_2, X_1)

Thats all I got so far, any hints

 

[Billy Bob]

Aren't you done? Isn't that what a^T \Sigma a is?

 

[cse63146]

Thought there was more to it than that.

There's another part of the question that says: Using Var (a_1 X_1 + a_2 X_2) show that for every choice of a1 and a2 that a^T \Sigma a \geq 0

Can I assume that \Sigma is always positive?

 

[Billy Bob]

Var (a_1 X_1 + a_2 X_2)\ge0 always, since it's variance! And you just showed it equals a^T \Sigma a