Having trouble understaning the notation co-variances

  • Thread starter Rabolisk
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In summary, the formula for covariance states that it is equal to the product of the standard deviations of two random variables, multiplied by the cosine of the angle between them. This formula holds true under the normal OLS assumption, and an extra 1/N term is needed if X and Y are matrices. There is also a close analogy between covariance and the cosine term in the formula for vector lengths.
  • #1
Rabolisk
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Can someone explain to me how
Cov(x,y)=[X-E(x))(Y-E(y)]

and this would equal 0 under the normal OLS assumption.


I know how to calculate the covariance, but X here is a matrix. So I don't understand the logic of this formula...
 
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  • #2
Welcome to PF, Rabolisk! :smile:

Did you already look at the wiki page?
http://en.wikipedia.org/wiki/Covariance

Your formula does not seem to be quite right.
X and Y could be random variables, but in that case an extra E should be present in your formula (see wiki).

If your X and Y are matrices then your formula should have an extra 1/N in it (see wiki).
 
  • #3
There is an interesting analogy between lengths-of-vectors and the variances-of-random-variables.

For vectors X and Y, we have
[itex] | X + Y |^2 = |X|^2 + |Y|^2 - 2|X||Y| \cos \theta [/itex] where [itex] \theta [/itex] is the angle between the vectors.

For random variables X and Y we have

variance(X+Y) = variance(X) + variance(Y) + 2 Covariance(X,Y)

The analogy is even better if we express it in standard deviations:

[itex] ( std. dev(X+Y))^2 = (std. dev(X))^2 + (std. dev(Y))^2 + 2 Covariance(X,Y) [/itex]

The covariance is roughly analagous to the cosine term. The quantity that would be analagous to [itex] cos(\theta) [/itex] is [itex] \frac{- Covariance(X,Y) }{(std. dev(X)) (std. dev(Y))} [/itex].
 

1. What is the purpose of using notation for co-variances?

The notation for co-variances helps to represent the relationship between two variables and their variability. It allows for a concise and standardized way of communicating this relationship in mathematical equations and statistical analyses.

2. How do you read and interpret notation for co-variances?

The notation for co-variances typically includes the two variables being analyzed, with a subscript indicating which variable is being measured. For example, if X and Y are the two variables, the co-variance notation would be represented as cov(X,Y). The value of the co-variance can then be interpreted as a measure of the direction and strength of the relationship between the two variables.

3. Is there a difference between co-variance and correlation?

Yes, there is a difference between co-variance and correlation. Co-variance measures the relationship between two variables, while correlation measures the strength of that relationship. Co-variance is also affected by the scale of the variables, whereas correlation is not.

4. How is the co-variance calculated?

The co-variance is calculated by taking the product of the difference between each pair of values for the two variables, and then finding the average of these products. This can be represented in the equation cov(X,Y) = (sum of (X - mean of X)(Y - mean of Y)) / n, where n is the number of data points.

5. Can co-variance be negative?

Yes, the co-variance can be negative. A negative co-variance indicates an inverse relationship between the two variables, meaning that as one variable increases, the other decreases. A positive co-variance indicates a direct relationship, where both variables increase or decrease together.

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