How Does the Covariance Matrix Apply to Vectors X and Y?

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Discussion Overview

The discussion revolves around the application of the covariance matrix to vectors X and Y, specifically focusing on the calculation of the covariance matrix and the differences between covariance definitions for random variables and random vectors. The scope includes theoretical aspects of covariance in statistics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that the covariance matrix for two variables X and Y should be a 2x2 matrix, with variances on the diagonal and covariances off-diagonal.
  • There is a question about whether cov(X,Y) and cov(Y,X) are considered off-diagonal elements in the covariance matrix.
  • Participants provide definitions of covariance for both real-valued random variables and random vectors, highlighting the mathematical expressions involved.
  • There is a query regarding the meaning of "dimensional" in the context of random vectors, with some suggesting it may refer to the number of samples.
  • One participant clarifies that the first definition refers to one-dimensional random variables, while the second is a generalization to multi-dimensional random vectors.

Areas of Agreement / Disagreement

Participants generally agree on the structure of the covariance matrix but express differing views on the interpretation of dimensionality and the definitions of covariance for variables versus vectors. The discussion remains unresolved regarding the implications of these definitions.

Contextual Notes

There are limitations regarding the assumptions made about dimensionality and the definitions of covariance, which may depend on the context of the variables involved. The discussion does not resolve these ambiguities.

DUET
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if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?
 
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DUET said:
if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?

You have two variables, so the matrix is 2x2. The elements are var(X), var(Y) along the diagonal and cov(X,Y) off diagonal (both).
 
Since 2x2 we need two diagonal elements and two off diagonal elements.

Are the following two elements "off diagonal elements"?

cov(X,Y) & cov(Y,X);
 
Last edited:
Yes they are the off diagonal elements.
 
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?
 
Last edited:
DUET said:
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?


In this context what do you mean by dimensional? X and Y are real valued. Do you mean the number of samples?
 
DUET said:
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?


1 refers to real valued (1 dimensional) random variables.
2 is a generalization to vectors (n or m dimensional) which have random variables as components.
 

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