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Cov(W,Z) where W=X and Z = aX+Y

  1. Oct 10, 2014 #1
    1. The problem statement, all variables and given/known data
    If ##X## and ##Y## have a covariance of ##cov(X, Y)##, we can transform them to a new pair of random variables whose covariance is zero. To do so, we let
    \begin{align*}
    W &= X\\
    Z &= aX + Y
    \end{align*}
    where ##a = -cov(X, Y)/var(X)##. Show that ##cov(W, Z) = 0##.

    2. Relevant equations
    ##var(X) = E[X^2] - E^2[X]##
    ##cov(X, Y) = E[XY] - E[X]E[Y]##

    3. The attempt at a solution
    SOLVED

    \begin{align*}
    cov(W, Z) &= E[(W - E[W])(Z - E[Z])]\\
    &= E\big[WZ - ZE[W] - WE[Z] + E[W]E[Z]\big]\\
    &= E[aX^2 + XY] - E[X]E[aX + Y]\\
    &= aE[X^2] + E[XY] - aE^2[X] - E[X]E[Y]\\
    &= \frac{-cov(X, Y)}{var(X)}E[X^2] +
    \frac{cov(X, Y)}{var(x)}E^2[X] + E[XY] - E[X]E[Y]\\
    &= cov(X, Y)\bigg[\frac{E[X^2] + E^2[X]}{E[X^2] - E^2[X]}
    + 1\bigg]
    \end{align*}

    $$
    avar(X) + cov(X, Y) = aE[X^2] + E[XY] - aE^2[X] - E[X]E[Y]
    $$
    So I found the error
     
    Last edited: Oct 10, 2014
  2. jcsd
  3. Oct 10, 2014 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    It is often easier and faster to use the well-known result
    [tex] \text{Cov}(S,T) = E(ST) - (ES)(ET) [/tex]
    for any random variables ##S,T## having finite first and second moments.
     
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