1. Suppose that y, x, z vectors in $\displaystyle \mathbb{R}^n$ where n > 3.Assume that the triplets (yi, xi, zi) 2 R3 are distinct. Suppose that Y,X, and Z are discrete random variables with joint probability mass function given by

$\displaystyle P((Y,X,Z) = (y_i, x_i, z_i)) = \frac{1}{n} $ for i = 1, 2, . . . , n

I would like to show that the two definitions are equal $\displaystyle \hat{p}_{yx.z} = p_{XY.Z} $ where definition 1 = definition 2

Definition 1: For a vector w, let $\displaystyle \tilde{w} = w - \bar{w}1$ where $\displaystyle \bar{w} = \frac{1}{n} \sum_{i=1}^n w_i $ to have a mean of 0. Then

$\displaystyle \hat{p}_{vw} = cor(v,w) = \frac{\tilde{v}' \tilde{w}}{\sqrt{(\tilde{v}' \tilde{v}) (\tilde{w}' \tilde{w})}} $

and

$\displaystyle \hat{p}_{vx.z} = cor \left( \tilde{y} - \frac{\tilde{y}' \tilde{z}}{ \tilde{z}' \tilde{z}} \tilde{z} , \tilde{x} - \frac{\tilde{x}' \tilde{z}}{ \tilde{z}' \tilde{z}} \tilde{z}\right) $ (1)

Definition 2:

$\displaystyle p_{YX.Z} $ $\displaystyle = cor \left( \tilde{Y} - \frac{E( \tilde{Y} \tilde{Z} )}{E( \tilde{Z} \tilde{Z})} \tilde{Z}, \tilde{X} - \frac{E( \tilde{X} \tilde{Z} )}{E( \tilde{Z} \tilde{Z})} \tilde{Z} \tilde{Z}\right) $ (2)

where E(X) is the expected value.

2. Suppose that for some large but fixed value of n, the n-vectors

x, z are fixed and have positive length ( x'x > 0, z'z > 0) and means 0 (that is, x'1 = z'1 = 0) and that for some constants a, b > 0 $\displaystyle y = bx + az $. Show that

[tex] \hat{p}_{yx.z} = 1

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For problem 1, let the joint pmf be defined as

$\displaystyle P((Y,X,Z) = (y_i, x_i, z_i)) = \frac{1}{n} $ for i = 1, 2, . . . , n (*)

I would like to go from show that definition 2 = definition 1.

At first glance, since $\displaystyle P(Y,X,Z) = \frac{1}{n} $ , then this can be defined as the uniform distribution. $\displaystyle p(X = x) = \frac{1}{b-a}$, where we can replace 'b' by n and 'a' by 0.

For the uniform distribution since the expected value $\displaystyle E[X] = \frac{a+b}{2}$ then the expected value of (*) is $\displaystyle \frac{n}{2} $

I know that for two centered random variables,

$\displaystyle E( \tilde{V} \tilde{W}) = E \left( E(V - E(V)) (W- E(W)) \right) = Cov(V,W) $

where $\displaystyle p_{VW} = Cor(V,W) = \frac{E( \tilde{V} \tilde{W} }{ \sqrt{E ( \tilde{V}^2) (E(\tilde{W}^2)}}$

but I am running around in circles.

Thank you for reading. Any help is greatly appreciated.