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A question about statistics

  1. Dec 15, 2012 #1
    1. The problem statement, all variables and given/known data

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

    a) Write u in terms of [tex]\rho[/tex] = the correlation coefficient, and [tex]\sigma[/tex]

    b) Find the distribution of t

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

    2. Relevant equations



    3. The attempt at a solution


    a) [tex]\rho = u/\sigma[/tex] so [tex] u = \rho\sigma[/tex]

    b) T is also normal...

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

    Let A=[tex]\begin{bmatrix} a & a-1\end{bmatrix}[/tex]

    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
     
  2. jcsd
  3. Dec 15, 2012 #2

    Ray Vickson

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    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.
     
  4. Dec 15, 2012 #3
    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?
     
  5. Dec 15, 2012 #4

    Ray Vickson

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    You cannot minimize Var(T) until you have a formula for Var(T). As I said, just use standard results to get it.
     
  6. Dec 16, 2012 #5
    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?
     
  7. Dec 16, 2012 #6

    Ray Vickson

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    How do you usually find maxima or minima of functions?
     
  8. Dec 16, 2012 #7
    I set the derivative equal to zero and then I check if those points are maximum or minimum...
     
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