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Math Help - Vector of independent normal RV's - Very difficult for me

  1. #1
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    Vector of independent normal RV's - Very difficult for me

    Let X = (X_1,...,X_n)^T be a column vector of independent N(0,1) distributed random variables. Let A be an n \times n orthogonal matrix.

    (1) Consider the random column vector Y = (Y_1,..., Y_n)^T, obtained via the linear transformation Y = AX. Show that Y is again a vector of independent N(0,1) distributed random variables.

    (2) Let A be an orthogonal matrix whose last row is (1....,1)/ \sqrt{n} . Consider again the linear transformation Y = AX. Show that

    W = \sum_{i=1}^n (X_i - \bar{X})^2 = \sum_{i=1}^n X_i^2 - n \bar{X^2} = \sum_{i=1}^n Y_i^2 \sim \mathcal{X}_{n-1}^2

    (3) Let Z_1, ..., Z_n be independent N ( \mu, \sigma^2) random variables, with sample variance S^2 = \frac{1}{n-1} \sum_{i=1}^n (Z_i - \bar{Z})^2. Show using (1) - (2) that (n-1)S^2 / \sigma^2 \sim \mathcal{X}_{n-1}^2
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by RoyalFlush View Post
    Let X = (X_1,...,X_n)^T be a column vector of independent N(0,1) distributed random variables. Let A be an n \times n orthogonal matrix.

    (1) Consider the random column vector Y = (Y_1,..., Y_n)^T, obtained via the linear transformation Y = AX. Show that Y is again a vector of independent N(0,1) distributed random variables.
    For this it is sufficient to show that the covariance matrix of Y is diagonal (and you can do that by considering what the i,j -th off-diagonal element must be).

    You will need to know some properties of orthogonal matrices, and how to form the covariance matrix of Y=AX, form A and the covariance matrix of X.

    CB
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  3. #3
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    Okay, I have been able to work out question 1. But I am really stuck on question 2 and consequently question 3.

    Can anyone please show me how answer question 2?
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