What is the significance of variance and covariance equations?

[dexterdev]

I have idea of physical signif of var(x),cov(x) but can't get derivations of equation.

Hi all,
I understood the facts that variance indicate the spread in random variable and covariance shows correlation between 2 r.v s etc. But I cannot imagine how we are arriving at their equations. Also what is the significance of nth moment etc?

TIA

-Devanand T

 

[chiro]

Hey dexterdev.

You can relate moments with the Fourier Transform and thus make an intuitive connection between the PDF and its frequency characteristics with moments (I'm not talking about central moments, just the standard ones).

With regards to the variance of multiple random variables, the real key to this is to look at it in the context of linear algebra with matrices rather than as an equation.

In multiple-dimensions, you have a covariance matrix and if you want to find the variance of a linear combination of variables, you are going to apply your covariance matrix to that vector just like you multiply a matrix and vector using Ax = b.

In the covariance instance, your x vector represents the vector that is a linear combination of the random variables (for example [3 4 5] would represent 3X1 + 4X2 + 5X3) and if A is the covariance matrix, then Var(X) = XAX^T where A is your covariance matrix and X is your vector of random variables.

If there is no covariance terms you get a diagonal matrix.

Now you must consider the nature of variance: it acts in some ways like a metric or norm and you are dealing with issues involving positive definite attributes and constraints that things like metrics and norms face when you look at them abstractly in higher dimensions.

These are the basics of the ideas but if you want more context you will have to dig deeper.

 

[ImaLooser]

Hi all,
I understood the facts that variance indicate the spread in random variable and covariance shows correlation between 2 r.v s etc. But I cannot imagine how we are arriving at their equations. Also what is the significance of nth moment etc?

TIA

-Devanand T

Variance is somewhat arbitrary. It works pretty well and is mathematically easy to work with.