Is there a more efficient way to do this?

[Artusartos]

Homework Statement

Let X= $$\begin{bmatrix} X_1 \\X_2 \end{bmatrix}$$ be bivariate normal $$N(\begin{pmatrix} \theta \\ \theta \end{pmatrix}, \begin{pmatrix} 1 & u \\ u & \sigma^2 \end{pmatrix})$$. Let $$T=aX_1 + (1-a)X_2$$.

a) Write u in terms of $$\rho$$ = the correlation coefficient, and $$\sigma$$

b) Find the distribution of t

c) Find the value of a that minimized the variance of T.

Homework Equations

The Attempt at a Solution

a) $$\rho = u/\sigma$$ so $$u = \rho\sigma$$

b) T is also normal...

If we write $$T = aX_1 + (a-1)X_2 = \begin{bmatrix} a & a-1\end{bmatrix}\begin{bmatrix} X_1 \\ X_2 \end{bmatrix}$$

Let A=$$\begin{bmatrix} a & a-1\end{bmatrix}$$

Then E(T)=AE(X)

Var(T) = A*covariance matrix*A'

So the distribution of T is N(E(T), Var(T))

Do you think this is correct?

c) I'm stuck here...


Thanks in advance

 

[Ray Vickson]

Homework Statement

Let X= $$\begin{bmatrix} X_1 \\X_2 \end{bmatrix}$$ be bivariate normal $$N(\begin{pmatrix} \theta \\ \theta \end{pmatrix}, \begin{pmatrix} 1 & u \\ u & \sigma^2 \end{pmatrix})$$. Let $$T=aX_1 + (1-a)X_2$$.

a) Write u in terms of $$\rho$$ = the correlation coefficient, and $$\sigma$$

b) Find the distribution of t

c) Find the value of a that minimized the variance of T.

Homework Equations

The Attempt at a Solution

a) $$\rho = u/\sigma$$ so $$u = \rho\sigma$$

b) T is also normal...

If we write $$T = aX_1 + (a-1)X_2 = \begin{bmatrix} a & a-1\end{bmatrix}\begin{bmatrix} X_1 \\ X_2 \end{bmatrix}$$

Let A=$$\begin{bmatrix} a & a-1\end{bmatrix}$$

Then E(T)=AE(X)

Var(T) = A*covariance matrix*A'

So the distribution of T is N(E(T), Var(T))

Do you think this is correct?

c) I'm stuck here...


Thanks in advance

You need to be careful: you need 1-a, not a-1.

Anyway, you know the mean of T and can get the variance using standard formulas, so you know the distribution of T without further work. You are doing it the hard way, and I do not understand what you are trying to do.

 

[Artusartos]

You need to be careful: you need 1-a, not a-1.

Anyway, you know the mean of T and can get the variance using standard formulas, so you know the distribution of T without further work. You are doing it the hard way, and I do not understand what you are trying to do.

Thanks for pointing out that mistake...

The reason why I'm trying to find E(T) and Var(T) is so that I can write it as N(E(T), Var(T)). I know that it's normal but I'm trying to find the mean and the variance so I can write it in this form. I'm not sure if the professor wants this so I'm just doing it...just in case.

But is it ok if you give me a hint for part c?

 

[Ray Vickson]

Thanks for pointing out that mistake...

The reason why I'm trying to find E(T) and Var(T) is so that I can write it as N(E(T), Var(T)). I know that it's normal but I'm trying to find the mean and the variance so I can write it in this form. I'm not sure if the professor wants this so I'm just doing it...just in case.

But is it ok if you give me a hint for part c?

You cannot minimize Var(T) until you have a formula for Var(T). As I said, just use standard results to get it.

 

[Artusartos]

You cannot minimize Var(T) until you have a formula for Var(T). As I said, just use standard results to get it.

After I computed Var(T) from Var(T) = A*covariance matrix*A', I set the derivative (with respect to a) to zero and solved for a...do you think that's correct?

 

[Ray Vickson]

After I computed Var(T) from Var(T) = A*covariance matrix*A', I set the derivative (with respect to a) to zero and solved for a...do you think that's correct?

How do you usually find maxima or minima of functions?

 

[Artusartos]

How do you usually find maxima or minima of functions?

I set the derivative equal to zero and then I check if those points are maximum or minimum...