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Having trouble understaning the notation co-variances

  1. Oct 28, 2011 #1
    Can someone explain to me how

    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...
  2. jcsd
  3. Oct 28, 2011 #2

    I like Serena

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    Homework Helper

    Welcome to PF, Rabolisk! :smile:

    Did you already look at the wiki page?

    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).
  4. Oct 29, 2011 #3

    Stephen Tashi

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    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].
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