1. Suppose that y, x, z vectors in 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

for i = 1, 2, . . . , n

I would like to show that the two definitions are equal where definition 1 = definition 2

Definition 1: For a vector w, let where to have a mean of 0. Then

and

(1)

Definition 2:

(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 . Show that

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

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For problem 1, let the joint pmf be defined as

for i = 1, 2, . . . , n (*)

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

At first glance, since , then this can be defined as the uniform distribution. , where we can replace 'b' by n and 'a' by 0.

For the uniform distribution since the expected value then the expected value of (*) is

I know that for two centered random variables,

where

but I am running around in circles.

Thank you for reading. Any help is greatly appreciated.