Covariance Matrix

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

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

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);

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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?

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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?

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.